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Research Collection
Doctoral Thesis
Multivariate extremes and regular variation for stochasticprocesses
Author(s): Lindskog, Filip
Publication Date: 2004
Permanent Link: https://doi.org/10.3929/ethz-a-004669275
Rights / License: In Copyright - Non-Commercial Use Permitted
This page was generated automatically upon download from the ETH Zurich Research Collection. For moreinformation please consult the Terms of use.
ETH Library
Diss. ETH No. 15319
Multivariate Extremes and
Regular Variation for
Stochastic Processes
A dissertation submitted to the
SWISS FEDERAL INSTITUTE OF TECHNOLOGY
ZURICH
for the degree of
Doctor of Mathematics
presented by
FILIP LINDSKOG
MSc in Engineering Physics KTH
born 20.10.1975
citizen of Sweden
accepted on the recommendation of
Prof. Dr. P. Embrechts, examiner
Prof. Dr. F. Delbaen, co-examiner
2004
to my parents
Acknowledgement
First I want to thank my supervisor Paul Embrechts for having confi¬
dence in me and for his constant encouragement. I am most grateful
to him for giving me the time and freedom needed to pursue the some¬
times vague ideas I believed would lead to interesting results. I also
want to thank my friend and colleague Henrik Huit for our cooperation
during the past years. I have benefited greatly from our joint work and
discussions during Henrik's visits to ETH Zürich and my visits to the
Mathematical statistics division at KTH in Stockholm. A special thank
goes to Alexander McNeil and Uwe Schmock for fruitful discussions and
joint work in various forms during the past years. Furthermore, I want
to thank Freddy Delbaen for being co-examiner and for his many inter¬
esting comments on the thesis. I am also very grateful to Credit Suisse,
Swiss Re and UBS for financial support through RiskLab (Switzerland)and for giving me the possibility to combine my work at RiskLab with
my PhD studies. Without this possibility this thesis could not have
been written. Finally I want to thank my colleagues at ETH Zürich for
their friendship and support.
v
Contents
Abstract ix
Kurzfassung xi
Introduction 1
1 Multivariate regular variation 9
1.1 Vague convergence 10
1.2 Vague convergence on the state space R \{0} 12
1.3 Multivariate regular variation 18
1.4 Sums of regularly varying random vectors 28
2 Regular variation for multivariate additive processes 35
2.1 Additive processes 38
2.2 Regular variation for multivariate additive processes and
for vectors of functionals acting on such processes ....41
3 Regular variation for general stochastic processes 65
vii
viii Contents
3.1 Regular variation on D 69
3.1.1 Proofs 73
3.2 Markov processes with asymptotically independent incre¬
ments 78
3.2.1 Proofs 85
3.3 Filtered Levy processes 91
4 Dependence in elliptical distributions 103
4.1 Elliptical distributions 104
4.2 Kendall's tau and Spearman's rho for elliptical distributions 109
4.3 Proof of Theorem 4.14 112
4.4 Proof of the counterexample 116
5 Multivariate extremes for elliptical distributions 125
5.1 The connection between regular variation and tail depen¬
dence 126
5.2 Interpretations and explicit computations of spectral mea¬
sures 131
Bibliography 138
Curriculum Vitae 143
Abstract
In this thesis extremes for random vectors and multivariate stochastic
processes are studied using the notion of multivariate regular variation
and its extensions to state spaces suitable for heavy tail analysis for
continuous-time stochastic processes. For multivariate stochastic pro¬
cesses X = (Xf)t>o questions concerning the implications of regular
variation of Xj for some fixed t > 0 on the tail behavior of various vec¬
tors of functionals acting on X are addressed. Formulations of regular
variation for the graph of a stochastic process and of regular variation
on the space of right-continuous functions with left limits are studied
and used to characterize the extremal behavior of stochastic processes
and to determine the tail behavior of vectors of functionals acting on
stochastic processes satisfying some of the regular variation conditions
considered. The formulation of regular variation on the space of right-
continuous functions with left limits is found to be particularly useful,
providing a powerful and elegant approach to heavy tail analysis for
stochastic processes similar to the classical weak convergence approach
of Billingsley [5].Another topic of this thesis is the study of various notions of dependence
and multivariate extremes for elliptical distributions. This includes the
concordance measures Kendall's tau and Spearman's rho, tail depen¬
dence coefficients and spectral measures with respect to different norms,
for which explicit expressions are computed.
ix
Kurzfassung
In dieser Dissertation werden Extreme von Zufallsvektoren und multi-
variaten stochastischen Prozessen untersucht. Hierbei benutzt wird das
Konzept der multivariaten regulären Variation und dessen Erweiterun¬
gen auf Zustandsräume die passend sind für die Analyse des asymp¬
totischen Verhaltens von stochastischen Prozessen in stetiger Zeit. Für
multivariate stochastische Prozesse X = (X)f>o wird untersucht, wie die
reguläre Variation von Xt für ein festes t > 0 das asymptotische Verhal¬
ten von verschiedenen Vektoren von Funktionalen beeinflusst, die auf X
operieren. Es werden Formulierungen studiert der regulären Variation
für den Graphen eines stochastischen Prozesses, sowie für rechtsstetige
Funktionen mit linksseitigen Grenzwerten. Diese werden benutzt, um
das extreme Verhalten von stochastischen Prozessen zu charakterisieren,
sowie um das asymptotische Verhalten zu bestimmen von Vektoren
von Funktionalen, die auf stochastischen Prozessen operieren, welche
einer bestimmten Form von regulärer Variation genügen. Die For¬
mulierung der regulären Variation auf dem Raum der rechtsstetigen
Funktionen mit linksseitigen Grenzwerten scheint besonders nützlich.
Sie ermöglicht einen mächtigen und eleganten Zugang zur asymptotis¬
chen Analyse von stochastischen Prozessen, ähnlich dem klassischen Zu¬
gang der schwachen Konvergenz von Billingsley [5].Ein weiteres Thema dieser Dissertation ist das Studium von diversen
Begriffen von Abhängigkeit und multivariaten Extreme von elliptischen
Verteilungen. Dies beinhaltet Kendalls tau, Spearmans rho, Koeffizien¬
ten der "tail dependence" und Spektralmasse bezüglich verschiedener
Normen, für welche explizite Ausdrücke berechnet werden.
xi
Introduction
The tail behavior of heavy-tailed random variables and functionals of
univariate stochastic processes has been studied successfully for quite
some time using the notion of regular variation. The theory of regularly
varying functions is an important field of real analysis with links to
number theory and complex analysis, but also to probability theory. If
X is a random variable with distribution function F, then X is said
to be regularly varying (at oo) with index a > 0 if 1 — F is regularly
varying (at oo) with index —a, i.e. if for every x > 0, as u —> oo,
l^FM1 - F{u)
Here X is considered heavy-tailed due to the relatively slow power-
law-type decay of 1 — F(u) as « —)• oo. Regular variation of X has
interesting implications for the tail behavior of various functions of X
and for the tail behavior of sums and maxima of independent copies of
X. All essential results on univariate regular variation can be found in
Bingham, Goldie and Teugels [7] ; a well written extensive collection of
results written with a high level of mathematical rigour. Various formu¬
lations of multivariate regular variation have been studied and used to
obtain results for stochastic phenomena where heavy tails appear in a
wide range of applications, including multivariate extreme value theory,
the description of weak limits of point process, the study of solutions to
stochastic recurrence equations, financial risk management and many
more. Here one is interested in the limit measure as n —> oo of a se¬
quence of measures of the form nP(a~1X <G • ), where X is a random
1
2 Introduction
vector and an —>• oo. Roughly speaking, the random vector X is said to
be (multivariate) regularly varying if the above sequence has a nonzero
limit measure. There are many other possible formulations of multivari¬
ate regular variation, but most of them have in common that they are
expressed in terms of convergence of measures. There is at this point
no similar collection of results for multivariate regular variation. Re¬
sults on multivariate regular variation appear in various places in more
or less different mathematical settings and although most basic results
on multivariate regular variation come as no surprise to those familiar
with the classical univariate theory, there are essential differences and
important results which do not seem to appear in the literature. There¬
fore, in Chapter 1 we give a detailed presentation of important aspects
of and fundamental results for multivariate regular variation. We also
derive results on linear transformations and sums of (multivariate) reg¬
ularly varying random vectors; these results are used extensively in the
subsequent analysis.
In applications one sometimes encounters data sets or time series with
a few extremely large observations. For such data sets it might be
appropriate to use heavy-tailed probability distributions to model the
underlying uncertainty. This is the case for instance in so-called catas¬
trophe insurance (fire, wind-storm, flooding) where the occurrence of
large claims may lead to large fluctuations in the cash-flow process
faced by the insurance company. The situation is similar in finance
where extremely large losses sometimes occur, indicating heavy tails of
the return distributions. The probability of extreme stock price move¬
ments has to be accounted for when analyzing the risk of a portfolio.
Another application is telecommunications networks where long service
times may result in large variability in the workload process. In many
applications it is appropriate to use a stochastic process (Xt)t>0 to
model the evolution of the quantity of interest over time. The notion
of heavy tails enters naturally in this context either as an assumption
on the marginals Xt or as an assumption on the increments Xt+h — Xt
of the process. However, it is often the case that the marginals or the
increments of the process are not the main concern, but rather some
functional of the process. A natural example is the supremum of the
process over a time interval, supir0T-| Xt. Another example is the av-
Introduction 3
erage of the process over a time interval, T_1 JQ Xtdt. We are then
typically interested in the probability that the functional exceeds some
high level, e.g. -What is the probability that the sea level exceeds a high
barrier sometime during [0,T]? It may therefore be important to know
how the tail behavior of the marginals Xt (or the increments) is related
to the tail behavior of functionals of the process. A lot of effort has been
put into studying the tail asymptotics of the distribution of functionals
of heavy-tailed univariate stochastic processes. Interesting results are
found in Embrechts, Goldie and Veraverbeke [15] under the assumption
of subexponentiality. For univariate infinitely divisible processes results
on the tail behavior for subadditive functionals are derived in Rosihski
and Samorodnitsky [36] also under assumptions of subexponentiality.
See also Braverman, Mikosch and Samorodnitsky [9] for further results
on the tail behavior of subadditive functionals of univariate regularly
varying Levy processes. However, whereas problems concerning the tail
behavior of univariate heavy-tailed stochastic processes (or functionals
of such) have been studied successfully for quite some time, multivariate
processes have received far less attention. Even though the intuition be¬
hind the univariate results to a large extent extends to the multivariate
case, proving results in the multivariate case requires other tools. The
reason for this is that the analytic notion of regularly varying or subex-
ponential functions which is used in univariate heavy tail analysis turns
out to be quite awkward and difficult to use when extended to the mul¬
tivariate case. In short, multivariate distribution functions are for many
purposes not natural objects causing the multivariate extensions to be
far from elegant and transparent. An approach based on convergence
of measures (described in detail in Chapter 1) makes the difference be¬
tween the univariate and multivariate heavy tail analysis disappear. It
also lends itself to nice geometrical interpretations and reduces tedious
technicalities to a minimum. For further discussions about this issue we
refer to Resnick [35] and the references therein. In the multivariate case
one typically studies a d-dimensional stochastic process (X.t)t>o- The
process could be interpreted for instance as measurements of sea levels
at d different locations, high-frequency return data of d different stocks
or claims in d different insurance lines. Clearly, an important difference
between the multivariate case and the univariate case when analyzing
4 Introduction
extremes is the possibility to have dependence between the components
of the random vector Xf. Large values may for instance tend to occur si¬
multaneously in the different components. One example is an important
macroeconomic event causing simultaneous big price drops for several
stocks. To have a good understanding of the dependence between ex¬
treme events in the multivariate case may be of great importance in
applications. Similar to the univariate case some functional or vector
of functionals of the process may be the primary concern. Natural
examples are for instance the componentwise suprema of the process
(supfGr0Tn JQ ,..., supfGr0)Ti X\ ). Another example is the compo¬
nentwise average of the process. Other functionals or combinations of
functionals may also be of interest. We are typically interested in the
probability that the vector of functionals belongs to some set far away
from the origin, e.g. -What is the probability that the sea level exceeds
a high barrier at some (or all) locations sometime during [0,T]? To
answer this type of questions we need to know how the tail behavior of
the marginals Xf is related to the tail behavior of vectors of functionals
of the process. This is the topic of Chapters 2 and 3.
In Chapter 2 we address these questions when the underlying stochastic
process is a multivariate additive process, i.e. a stochastically continuous
process with independent increments starting at zero. More precisely,
we consider an additive process (X.t)t>o and we study the implications
of Xf being regularly varying for some fixed t > 0 on the tail behavior of
vectors of functionals depending on the sample path of the process up to
time t. For example, we find that regular variation of Xf implies regular
variation of (supiGr0T] XJ; ,... ,suptGr0T] JQ ) with a limit measure
that is fully determined by that of X^. For functionals more complicated
then e.g. the componentwise suprema, knowing only the tail behavior
of X^ is insufficient for determining the tail behavior of the functionals
of the sample path up to time t. Therefore, we study a formulation
of regular variation for the graph of a multivariate stochastic process
and we give necessary and sufficient conditions for an additive process
to have a regularly varying graph. We then use this formulation for
deriving sufficient conditions for regular variation of the random vector
(J0 Xs ds,..., J*0 Xs ds) when (Xs)sG[0,i] is a d-dimensional additive
process with a regularly varying graph, and we show how the respective
Introduction 5
regular variation limit measures are related.
In Chapter 3 we consider a different and more general approach to
heavy tail analysis for stochastic processes. Just as multivariate regu¬
lar variation provides a natural way for understanding the tail behav¬
ior of heavy-tailed random vectors, a similar formulation for stochastic
processes with sample paths in D([0,l],Md) - the space of Md-valued
right-continuous functions on [0,1] with left limits - provides us with a
natural and powerful framework for studying the extremal behavior of
heavy-tailed stochastic processes. The chapter can be divided into three
main parts. In the first part we give two formulations of regular varia¬
tion on .D([0,1], Md) and we show that they are equivalent. We then give
necessary and sufficient conditions for regular variation on D([0,1], Md)in terms of multivariate regular variation of the finite-dimensional dis¬
tributions and relative compactness. We also give a version of the Con¬
tinuous Mapping Theorem for this setting, a result which proves very
useful in the subsequent analysis. In the second part we consider strong
Markov processes (not necessarily time-homogeneous) with asymptoti¬
cally independent increments in a certain sense. We show that for such
processes the sufficient conditions for regular variation on D([0, l],Md)simplifies considerably; they are almost the same as the necessary and
sufficient conditions for having a regularly varying graph. Moreover, we
find that if a strong Markov process with asymptotically independent
increments and sample paths in D([0, l],Ed) is regularly varying, then
its regular variation limit measure concentrates on step functions with
one step. This means that the process hits a Borel set in Mr" far away
from the origin by making one big jump and, in comparison to the size
of the jump, does not move much before and after the jump. More¬
over, knowing that the regular variation limit measure concentrates on
step functions with one step we can explicitly compute the regular vari¬
ation limit measure of many interesting vectors of functionals acting
on the process by simply applying our version of the Continuous Map¬
ping Theorem. In the third part we consider processes Y which can
be expressed as an integral of a deterministic function / : [0, l]2 —> R
with respect to a regularly varying Levy process X. The idea here is
that we can compute the regular variation limit measure of h(X), where
h : D([0, l],Ed) -)- D([0, l],Rd) satisfies the conditions of the Continu-
6 Introduction
ous Mapping Theorem enabling us to derive the extremal behavior of
h(K) from that of X and the properties of /, and show that Y and
h(K) have the same regular variation limit measure. To exemplify this
idea, we explicitly compute the regular variation limit measure of an
Ornstein-Uhlenbeck process driven by a regularly varying Levy process.
We find that the framework set up and studied in this chapter is general
enough to apply to the majority of interesting problems that arise in
multivariate heavy tail analysis for stochastic processes, yet powerful
enough to produce explicit results.
The canonical example of a multivariate distribution with a nontrivial
dependence structure is the multivariate normal distribution. A large
number of multivariate models, in particular in mathematical finance,
are based on the multivariate normal distribution since it enables ef¬
ficient simulation and explicit computation of many interesting quan¬
tities. However, at a closer look one often finds that the multivariate
normal distribution and models based on it do not capture essential
properties indicated by the observed data. For example, the empiri¬
cal distribution of financial return data has typically much heavier tails
than the normal distribution. Moreover, there is a tendency of simul¬
taneous large negative returns indicating a strong dependence between
extreme returns which cannot by properly modelled by a multivariate
normal distribution. In Chapter 4 we study the most natural extension
of the multivariate normal distribution, namely the class of elliptical dis¬
tributions. The class of elliptical distributions provides a rich source of
multivariate distributions which share many of the tractable properties
of the multivariate normal distribution and enables modelling of joint
extremes and other forms of nonnormal dependences. We show how
one can use elliptical distributions to construct multivariate discrete
time stochastic processes having elliptical finite-dimensional distribu¬
tions. Moreover, we study the concordance measures Kendall's tau and
Spearman's rho which play an important role in parameter estimation
for models based on elliptical distributions. In Chapter 5 we analyze
multivariate extremes for elliptical distributions. We find that the gen¬
eral stochastic representation of elliptically distributed random vectors
enables us to explicitly compute interesting quantities such as coeffi¬
cients of tail dependence (which measure the strength of dependence of
Introduction 7
joint extremes) and spectral measures associated with regularly vary¬
ing random vectors (a spectral measure is a probability measure on the
unit sphere with respect to some norm; the probability assigned to a set
represents the likelihood of an extreme observation being located in the
corresponding directions). The analysis in this chapter also highlights
various aspects of the concept of multivariate regular variation intro¬
duced in Chapter 1. The explicit expressions for densities of spectral
measures derived offer possibilities to study the effect on the depen¬
dence of multivariate extremes when varying the correlation structure
and the tail index of the marginals. Moreover, we study the effect of
different choices of norms - corresponding to different criteria for what
constitutes an extreme multivariate observation - on how the spectral
measure assigns probability mass to different directions. For example,
X may be considered extreme either if |X|2 = \/XTX exceeds some
high threshold, or if |X|oo = maxj \X^\ exceeds some high threshold.
Comments on the thesis
The first three chapters of this thesis are based on the papers [23] and
[24] (with H. Huit). The last two chapters which are based on the papers
[28] (with A. McNeil and U. Schmock) and [22] (with H. Huit) are less
technical and represent parts of my work within RiskLab. Although
the thesis is based on the papers mentioned above many new results,
comments and examples have been added to the thesis. The material
presented in the first three chapters is structured and presented as it is
with the aim of presenting a general framework for heavy tail analysis
for stochastic processes as well as a collection of relevant and useful
results on this topic. Chapter 3 can be considered as the main chapter.
However, in order to fully appreciate the usefulness of the approach and
the results presented in Chapter 3 a comparison with the approach and
analysis of Chapter 2 is helpful.
Throughout this thesis we assume as given a probability space (Q, J7, P)on which all random elements are defined.
Chapter 1
Multivariate regular
variation
In this chapter we introduce the notion of multivariate regular variation
and give results for regularly varying random vectors and for sums of
such. The chapter is organized as follows. In Section 1.1 we review the
notion of vague convergence following Kallenberg [26] and Resnick [34].In Section 1.2 we focus on properties of vague convergence on state
spaces relevant for multivariate regular variation. In particular we give
sufficient conditions for vague convergence on these state spaces which
will be used in the subsequent chapters. In Section 1.3 we introduce the
notion of multivariate regular variation. In particular we give impor¬
tant properties of the limit measure associated with a regularly varying
random vector and we give an equivalent formulation of multivariate
regular variation which will facilitate interpretations of results derived
in the following chapters. In Section 1.4 we give results on the tail
behavior of sums of regularly varying random vectors. These results
are of independent interest but, more importantly, will play an impor¬
tant role in the following chapters when we study regular variation for
multivariate stochastic processes.
9
10 Chapter 1. Multivariate regular variation
1.1 Vague convergence
Let (.Ë, p) be a locally compact, complete and separable metric space
and let B(E) be the Borel a-algebra on E generated by the p-open sets.
For B C E we denote by B°,B and dB = B\B° its interior, closure
and boundary, respectively. If B C E, then (B, p) is a metric space and
B(B) = B(E)r\B,
where B(E)nB = {AnB :Ae B{E)} (see e.g. p. 224 in Billingsley [5]).A set B C E is said to be bounded or relatively compact if its closure
B is compact. Let C^-(E) denote the class of all continuous functions
/ : E —)• K_|_ = [0,oo) with compact support, i.e. if / G C^-(E), then
there exists a compact set K such that f(x) =0 for x G Kc. Let M+(E)be the class of all Radon measures on (E, B(E)), i.e. of all (nonnegative)measures fi such that fi(B) < oo for all relatively compact B G B(E).In order to discuss convergence of Radon measures on (E,B(E)) we
must topologize M+(E). The class of all finite intersections of M+(E)-sets of the form {/i G M+(E) : fE f(x)p(dx) G (s,t)} with arbitrary
/ G Ck(E) and s,t G M form a base for a topology on M+(E) which
is called the vague topology. The space M+(E) with this topology can
be metrized as a complete and separable metric space, see e.g. Kallen¬
berg [26] for an example of a possible metric. We note that a sequence
(/in), fin G M+(E), converges to fi G M+(E) in the vague topology,
written /j,n A- //, if and only if JE f{x)fin{dx) —> JE f(x)fi(dx) for ev¬
ery / G Ck(E). When considering the subspace consisting of all finite
measures on M+(E), then we obtain the so-called weak topology by
replacing C^-(E) by the class of all bounded and continuous functions
f : E —> K_|_. Similarly, (fin) converges to /i in this topology, written
pn A- /i, if and only if JE f{x)fin{dx) —> JE f(x)/i(dx) for every nonneg¬
ative, bounded and continuous /. A theoretical justification for looking
at sets rather than integrals is given by the following result.
Theorem 1.1 (Kallenberg [26], 15.7.2) Let p, /ii, fi2, • • •be Radon mea¬
sures on (E,B(E)). Then the following statements are equivalent.
(l) fln A fl,
1.1. Vague convergence 11
(ii) pn(B) —> p(B) for every relatively compact B G B(E) such that
p{dB) = 0.
(iii) limsupn^00 pn(F) < p(F) and liminfj^oo pn(G) > p(G) for ev¬
ery compact F G B(E) and every open relatively compact G G
B(E).
Although studying integrals rather than sets may lead to a sometimes
more elegant analytic approach, we prefer the latter since this leads to
more intuitive geometric interpretations.
Remark 1.2 Note that \imsup pn(F) < p(F) for every compact F if
and only if liminf pn(G) > p(G) for every open relatively compact G.
This can be shown as follows. Suppose that \imsup pn(F) < p,(F) for
every compact F. Take an arbitrary open relatively compact G. Then,
for some e > 0, D = {x G E : p(x, G) < e} is compact, p(dD) = 0 and
p(G) = p(D) - p(D\G) < \im pn(D) - limsup pn{D\G)n
n
< nmpn{D) - liminf pn{D\G) = liminf(/*„(£>) - pn{D\G))n n n
= liminf pn{G).n
Conversely, suppose that liminf pn{G) > p(G) for every open relatively
compact G. Take an arbitrary compact F. Then, for some e > 0,
D = {x G E : p(x, F) < e} is open and relatively compact, p(dD) = 0
and
p{F) = p(D)-p(D\F)>\impn{D)-\immîpn{D\F)n n
> \im pn(D) - lim sup pn(D\F) = limsup(/in(D) - ^n(D\F))n
n n
= \rmsup pn(F).n
Remark 1.3 If p,pi,p2, • • • are finite measures on (E,B(E)), then
pn A- p if and only if pn A- p and pn(E) —> p(E) (Kallenberg [26],
15.7.6).
12 Chapter 1. Multivariate regular variation
We will frequently show vague convergence for a sequence (pn), P-n £
M+(E), by showing that the set {pn} is relatively compact in the vague
topology on M+(E) and that any two subsequential vague limits coin¬
cide. Therefore, we will need the following result.
Theorem 1.4 (Kallenberg [26], 15.7.5) A subset II of M+(E) is
relatively compact in the vague topology if and only if
sup p{B) < oo
neu
for every relatively compact B G B(E).
1.2 Vague convergence on the state space
Wd\{o}
Let | • | be any norm on Md and let Vß denote the Borel u-algebra gen¬
erated by the open sets with respect to the metric |x — y|, i.e. 7ld
is the usual Borel cr-algebra on Md. For x G Md and e > 0, let
B*,e = {y £ ^d ' |x — y| < e}. For x G R, let \x\ denote the ab¬
solute value of x. We will study vague convergence of measures of the
form n P(a~1X G • ) as n —> oo, where X is an Revalued random vector
and 0 < an t oo. Clearly, for every e > 0, B0e is relatively compact in
Md but nP(a~1X G -B0,e) ~* °°- Moreover, for an appropriate choice of
(an) we would expect (nP(a~1X G Bq J) to converge, but since vaguely
convergent measures only ensure convergence for relatively compact sets
this can not be established with this topology. To solve this problem we
will modify the state space Md and equip the modified state space with a
topology which renders sets B G 7ld bounded away from 0 (0 ^ B) rel¬
atively compact. Multivariate regular variation is typically formulated
in terms of vague convergence of Radon measures on M \{0}, where
M = [oo, oo]. We will show that M \{0} can be equipped with a metric
which renders M \{0} a locally compact, complete and separable metric
space. Moreover, with this choice of metric
nd n (Md\{o}) = £(Rd\{o}) n (Md\{o}),
1.2. Vague convergence on the state space M \{0} 13
j Cl
i.e. on Ed\{0} the Borel cr-algebra B(M \{0}) coincides with the usual
Borel cr-algebra 7ld, and every B G lZd bounded away from 0 is relatively
compact. Note that the measures we will consider assign zero mass to
M \Md ; the points of M \Md are of no interest apart from being a part
of the modification of the state space which enables us to use the notion
of vague convergence.
Theorem 1.5 There exists a metric p such that
(i) (M \{0},/)) is a locally compact, complete and separable metric
space and
(ii) if B(M \{0}) denotes the Borel a-algebra generated by the ~p-open
sets, then Ud n (Rd\{0}) = #(Rd\{0}) n (Rd\{0}).
Proof. Let W\{0} = (0, oo] xS^1, where S^1 = {x G Md : [x^ = 1}denotes the unit sphere with respect to the max-norm | • |oo given by
|x|oo = max^a^1)],..., |#(d)|). For x G Rd\{0} we write x = (x*,x).
Equip Rd\{0} with the metric
p(x,y) = max(|l/z* - l/y*\, |x - y|).
Since (0, oo] is complete and separable with the metric \l/x* — 1/y*| and
since S^"1 is complete and separable with the metric |x — y| it follows
that Rd\{0} is complete and separable with the metric p. Moreover,
Rd\{0} is locally compact; for every x G Rd\{0} there exists an e > 0
such that Bx e= {y G Rd\{0} : p(x, y) < e} is compact. Let
T: (Rd\{0},ftdn(Rd\{0}))
- ((0, oo) x 8d^\B(W\{0}) n ((0, oo) x S^1))
be given by T(x) = (Ix^x/lx^). Then T is continuous, one-to-one
and onto and T~l given by T_1((£*,x)) = x*5i is continuous. Hence,
there is a one-to-one correspondence between the subspace topologies on
Rd\{0} and (0, oo) x S^"1. With the convention arctan(ioo) = ±tt/2
and tan(±7r/2) = ±oo, let / : Rd\{0} - W\{0} be given by
/(x) = (|x|00,5r(x)/|p(x)|00),
14 Chapter 1. Multivariate regular variation
where p(x) = (-| arctan^1)),..., ^ arctan(a;(d))). Then / is one-to-one
and onto and
/-1((£*, x)) = (tan(arctan(x*)x(-1-)),... ,tan(arctan(x*)x(-d-))).
Equip R \{0} with the metric /j(x,y) = p(/(x),/(y)). It is easily
shown that R \{0} is separable and locally compact. We now show
that it is complete. Let (xn) be a Cauchy sequence in R \{0}, i.e. (xn)satisfies p(f(xn), /(xm)) —y 0 as m, n —)• oo. Then (/(xn)) is a Cauchy
sequence in Rd\{0} and by completeness there exists z G Rd\{0} such
that p(/(xn),z) —)• 0 as n —> oo. Since / is onto there exists x such
that z = /(x) and hence p(f(xn), /(x)) —>• 0 as n —> oo. Hence, R \{0}is complete. By a similar argument one can show that / and /_1 are
continuous. Moreover, the restriction
/:(Rd\{0},^(Rd\{0})n(Rd\{0}))- ((0, oo) x s^1, B(W\{o}) n ((0, oo) x s^1))
of/ is continuous, one-to-one, onto and has a continuous inverse. Hence,
also T~lof is continuous, one-to-one, onto and has a continuous inverse.
It follows that
nd n (Rd\{o}) = £(Rd\{o}) n (Rd\{o}),
i.e. the Borel sets we are interested in are the usual ones. D
We call a subclass U oïB(E) a convergence-determining class if pn{B) —>•
p{B) for every /i-continuity set B G U implies pn A- p on B(E). We
now present some useful examples of convergence determining classes
for vague convergence on B(M \{0}).
d
Theorem 1.6 Let p, pi, P2, • • •be Radon measures on B(M \{0}) with
yu(Rd\Rd) = 0 and letU be a subclass ofUd n (Rd\{0}) such that
(i) U is closed under the formation of finite intersections and
(ii) each open set in Md bounded away from 0 is a finite or countable
union of elements ofU.
J
1.2. Vague convergence on the state space R \{0} 15
If pn(A) ->> p(A) for every AeU, then pn A p on B(M \{0}).
Proof. If Ai, ..., Am lie in U, then so do their intersections; hence, by
the inclusion-exclusion formula,
fj,n(UZiAi) = J2^(Ai)-J2^AinAj">i i<j
+ J2 fJ-niAiHAjHAk)-...i<j<k
-+ Y^KAiï-^KAinAj)i i<j
+ J2 KAinAjHAk)-...i<j<k
= K^ÏLiAi).
If G is open and relatively compact in R \{0}, then GnRd is open and
relatively compact in R \{0} and open and bounded away from 0 in
Md. Hence, GflRd = UiAi for some sequence (Ai) of elements of U and
p(G) = p(G n Md). Hence, for any e > 0, we can choose m such that
p(Ui<mAi) > p(G) - e. Thus,
liminf/iTl(Gf) > liminf pn(\Ji<mAi)n—^-oo n—>-oo
= p(yJi<mAi) > p(G) - e.
Since e was arbitrary, liminf^^oo pn(G) > p(G). The conclusion follows
from Theorem 1.1 and Remark 1.2. D
Remark 1.7 Note that Theorem 1.6 still holds if R and Md are re¬
placed by R+ and R^_.
We now present convergence determining classes that will prove useful
in the subsequent analysis. Typically we will prove convergence to a
limit measure p which we know has the scaling property of Theorem
1.8 below. The convergence determining classes presented below remain
convergence determining classes without the additional condition on the
16 Chapter 1. Multivariate regular variation
limit measure p, but to avoid unnecessary lengthy arguments we prove
the results below under this stronger assumption.
Let | • | be any norm on Md and let §d_1 = {x G Md : |x| = 1}. For
u > 0 and S G B^'1) let Vu,s = {x G Md : |x| > w,x/|x| G S}.
Theorem 1.8 If p is a Radon measure onB(M \{0}) with p(M \Md) =
0 such that for some a > 0 p(uB) = u~ap(B) for every u > 0 and
5 G #(Rd\{0}); tfien
(%) /i(wSd_1) = 0 for every u > 0,
(ii) /i({x}) = 0 for every x G R \{0};
(Hi) p(dVUjS) = p(Vu,ds) for every u > 0 and S G #(§d_1);
(^y) /^(Vo^s}) = 0 for all but at most countably many s G Sd~1.
Proof, (i) Suppose that there exist u,c > 0 such that p(uSd~1) = c.
Then, for v > u,
/i(Vu>s--i\V0>s«.-i) > /iHeQnc«,«]^-1) = ^ M«®*-1)çGQn(u,t;]
> c(v/u)-a J2 X = °°-
Since Fu gd-^V^ §d-i is relatively compact this is a contradiction and
we conclude that /z(n§d_1) = 0.
(ii)By(i),M({x})</i(|x|§d-1) = 0.
(iii) By (i) and since /i(Rd\Rd) = 0, p(dVu,s) = p(uS) + p(Vu,ds) =
MK,ös)-(iv) Since p is a Radon measure p(Vißd-i) < oo and hence p(Vi^sy) = 0
for all but at most countably many s G Sd~1. By the scaling property,
p(Vi^sy) = 0 if and only if p(Vu^sy) = 0 for every u > 0. Letting u 4- 0,
the conclusion follows. D
Theorem 1.9 Let p, pi, P2-, • • •be Radon measures on B(M \{0}) such
that p satisfies the conditions of Theorem 1.8. If V = {VUjs ' u > 0, S G
1.2. Vague convergence on the state space R \{0} 17
B(Sd-1),p(dVUjS) = 0} and if pn(A) -> p(A) for every A e V, then
pnA p onB(Md\{0}).
Proof. Let U0 = {VUtS\VVtS 0 < u < v,S G B^'1)}. Clearly
Uq satisfies the conditions of Theorem 1.6. By Theorem 1.8 also U =
{A G Uo : p(dA) = 0} satisfies the conditions of Theorem 1.6 (U con¬
tains those elements of Uo that remain after removing the elements
Vu,s\Vv,s f°r which VUjds has a nonempty intersection with one of the
at most countably many rays Vo,{5} charged by p). By Theorem 1.8,
Vu,s\Vv,s G U if and only if K,s, Vv,s G V and if //„(K,s) - MK,s)and /^(K,s) - m(K,s), then
Mn(K,s\K,s) = Vn(Vu,s) ~ Vn(Vv,s)
-+ KVu,s) - KVv,s) = v(Vu,s\Vv,s)-
Hence, if pn(A) —* p(A) for every A G V, then pn(A) —y p(A) for every
A eU which in turn implies that pn A p on B(M \{0}). D
For a, b G Md we write a < b if a^ < 6^^ for i = 1,..., d. For a < b we
write [a, b) = [a^\b^) x • • • x [a(d), b^). If /i satisfies the conditions
of Theorem 1.8 and if [a, b) is bounded away from 0, then by Theorem
1.8 (i) with | • | being the max-norm, p(d[a.,h)) = 0; i.e. all sets [a,b)bounded away from 0 are //-continuity sets. Clearly those sets also
satisfy the conditions of Theorem 1.6.
—dLemma 1.10 Let p: pi, P2, • • •
be Radon measures on B(M \{0}) such
that p satisfies the conditions of Theorem 1.8. Let U be the collection
of sets of the form [a, b) such that a, b G Rd\{0}7 a < b and [a, b) is
bounded away from 0. If pn(A) —> p(A) for every A EU, then pn A p
onB(Md\{0}).
Clearly one can replace the collection of such half-open intervals by some
arbitrary combination of open, closed and half-open intervals and the
resulting collection of sets will remain a convergence determining class
for vague convergence on B(M \{0}) to a limit measure of the type of
Theorem 1.8.
18 Chapter 1. Multivariate regular variation
Note that since R+\{0} is equipped with the subspace topology, the
boundary of elements in U with nonempty intersection with the coor¬
dinate hyperplanes have empty intersection with the coordinate hyper-
planes (for R+\{0} equipped with the subspace topology of R \{0} we
have for example ([1,2] x [0,1])° = (1,2) x [0,1)). Hence U always
contains the sets [a, b) which have nonempty intersections with the co¬
ordinate hyperplanes. Hence, if we would replace each set [a, b) in U
with (a, b), then we would no longer have a convergence determining
class.
—dLemma 1.11 Let p, pi, P2, • • •
be Radon measures on B(M+\{0}) such
that p satisfies the conditions of Theorem 1.8. Let U be the collection
of sets of the form [a, b) such that a, b G R+\{0} and a < b. //
pn(A) -t p(A) for every AeU, then pn A p on #(R+\{0}).
1.3 Multivariate regular variation
The notion of multivariate regularly varying random vectors has ap¬
peared in several apparently different applications such as the study of
stochastic recurrence equations (see Kesten [27]), multivariate extreme
value theory (see e.g. Resnick [34]), the study of domains of attraction
of multivariate distributions (see Rvaceva [37]) and the description of
weak limits of point processes constructed from stationary sequences
of random vectors (see Davis and Hsing [13]). In recent years some
effort has been made to establish equivalence of the different notions
of multivariate regular variation (see Basrak [2] and Basrak, Davis and
Mikosch [4]) and as a result several equivalent definitions exist. We will
consider mainly two of them. The following definition is perhaps the
most useful one.
Definition 1.12 An Md-valued random vector X is said to be regularly
varying if there exist a sequence (an), 0 < an "f oo and a nonzero Radon
1.3. Multivariate regular variation 19
measure p on B(M \{0}) with p(M \Md) = 0 such that, as n —y oo,
nP(a"1XG •) Ap(-) onB(Md\{0}). (1.1)
Remark 1.13 A standard regular variation argument shows that if
(1.1) holds, then the sequence (an) is regularly varying with index 1/a,i.e. for every A > 0, a^xn]/an —* ^l^a as n —> oo.
If Definition 1.12 holds, then the limit measure p has the following
scaling property which will play an important role in the subsequent
analysis.
Theorem 1.14 // the conditions of Definition 1.12 hold, then there
exists an a > 0 such that p(uB) = u~ap(B) for every u > 0 and
BeB(Md\{0}).
Proof. For r > 0 and S G #(Sd_1), let K,s = {x G Rd : |x| >
r, x/|x| G S}. Fix S G jB(8d_1) such that for some r > 0 we have
p(dVrjs) = 0. Since p is a Radon measure such a set S G #(§d_1) exists.
Then p(duVr,s) = 0 for all but at most countably many u G [l,oo).Denote this set by U, i.e. U = {u £ [1, oo) : p(duVr,s) = 0}.
Suppose that p(Vr,s) > 0. Define / and g on (0,oo) by f(x) = P(X G
xVr,s) and g(x) = p(xVrjs)- Then, for every u G U, as n —> oo,
nf(uan) = nP(X G anuVr,s) -> n(uVr,s) = g{u).
For x > ai, let t = t(x) be the largest integer with a* < x. Since / is
nonincreasing, i.e. x < y implies f(x) > f(y),
f(uat+1)/f(at) < f(ux)/f(x) < f(uat)/f(at+ï).
However, the lower bound is (t/(t + l))(t + l)f(uat+i)/(tf(at)) which
tends to g(u)/g(l), for each u G U. Similarly for the upper bound.
Hence, for every u G U, as x —y oo,
f(ux)/f(x)^g(u)/g(l).
20 Chapter 1. Multivariate regular variation
For arbitrary u > 0, let g*(u) = lim sup^.^^ f(ux)/f(x). Then, g*(u) =
g(u)/g(l) for u G U. Moreover, g*(u) < 1 for u > 1 since f(ux)/f(x) <
1 for u > 1 and x > 0. In particular, limsup^ g*(u) < 1. It now
follows by Theorem 1.4.3 (ii) p. 18 in Bingham, Goldie and Teugels [7]that there exists an a G R such that, for every u > 0, as x —y oo,
f(ux)/f(x)^u~a,
i.e. p(uVrjs) = u~ap(Vrjs) for every u > 0.
Suppose that //(V^s) = 0. We will show that this implies that p(Vu,s) =
0 for every u > 0. Suppose that there exists ro G (0,r) such that
^{Yr0,s) > 0. Then, by the above arguments, there exists r\ G (0,r)such that p(Vrijs) > 0 and p(dVrijs) = 0. However, we must then have
p(uVrijs) = u~ap(Vri^s) for every u > 0. In particular, 0 = p(Vr,s) =
(r/ri)~ap(Vri,s) > 0 which is a contradiction.
Hence, for each yit-continuity set Vrjs there exists an a G R such that,
for every u > 0, //(uV^s) = w_a/i("\/rj£').It remains to show that a does not depend on Vr,s- This can be shown
by the same arguments as in Resnick [34] p. 277. Since the /i-continuity
sets of the form Vr,s for r > 0 and S G B(E>d~l) form a 7r-system
which generates B(M \{0}) fl R the scaling property holds for every
B G #(Rd\{0})nRd. However, since //(Rd\Rd) = 0 the scaling property
holds for every B G B(M \{0}). Since p is a Radon measure we must
have a > 0 and since p(M \Md) = 0 we must have a > 0. D
The following equivalent formulation of multivariate regular variation
will provide many nice interpretations of various regular variation re¬
sults in the following chapters.
Theorem 1.15 Let X be an Md-valued random vector. Then the fol¬
lowing statements are equivalent.
(i) X is regularly varying in the sense of Definition 1.12.
(ii) There exist an a > 0 and a probability measure a on #(8d_1) such
that, for every x > 0, as u —y oo,
1.3. Multivariate regular variation 21
Remark 1.16 If (i) holds, then by Theorem 1.14 there exists an a > 0
such that p(xB) = x~ap(B) for every x > 0 and B G #(Rd\{0}) and
(ii) holds with the same a. If (ii) holds, then (i) holds and p satisfies
the scaling property above with the same a (see the proof of Theorem
1.15 for details).
Definition 1.17 For a random vector X satisfying (1.2) we refer to a
and a as the tail index of X and the spectral measure of X with respect
to the norm \ \, respectively.
An immediate consequence of Theorem 1.15 is the following equivalent
formulation of regular variation for a nonnegative random variable.
Corollary 1.18 A nonnegative random variable R is regularly varying
(at oo,) with index a > 0 if and only if for every x > 0,
..
M(R > ux)hm J.
„f- = x .
u^oo F(R > u)
Example 1.19 Let X = RTJ, where the nonnegative random variable
R and the §d_1-valued random vector U are independent. Suppose that
R is regularly varying with index a > 0. Then
P(|X|>^,X/|X|G-) ¥(R > ux)
P(|X| >u) ¥(R>u){ )
^y x-aF(Ue-)
on #(§d_1) as u —y oo, i.e. X is regularly varying with index a and
spectral measure P(U G • ) with respect to the norm | • |.
For more interesting (nontrivial) examples, see Chapter 5.
Proof of Theorem 1.15. (ii) => (i) For x > 0 and S G B^'1), let
VXtS = {xGRd :|x| >x,x/|x|eS},
V = {V^s'-xyO^SeB^-1)}.
22 Chapter 1. Multivariate regular variation
For u, x > 0 define the measures
Hu(Vx,) = P(|X|>«x,X/|X|G.)/P(|X|>«),
p(Vx,) = x'aa(-)
on #(§d_1). This also defines, uniquely, set functions pu and p on the
semiring V. By Theorem 11.3 p. 166 in Billingsley [6], pu and p can be
uniquely extended to measures on a(V) = IZd D (Rd\{0}). By requiring
that pu(M \Md) = p(M \Md) = 0, pu and p can be uniquely extended
to (Radon) measures on B(M \{0}). By definition of p, p(VXis) =
x~ap(Vi:s) for every x > 0 and S G #(§d_1). Suppose that there exist
x, c > 0 such that p.(dVx§d-i) = c. Then, for y > x,
M^s^-AK/,^-1) > KuqeQn(x,y]dVqjSd-i) = 22 M^V^d-i)qe<Qn(x,y]
> c(y/x)~a J2 1 = °°-
qeQn(x,y]
Since ^gd-i^gd-i is bounded in R \{0} this is a contradiction and
we conclude that p(dVx §d-i) = 0 for every x > 0. This implies in partic¬
ular that p(dVXjs) = p(Vx,ds) for every x > 0 and 5 G #(§d_1). Since
MuO^,-) -^ Kvx,-) for every a; > 0 this implies pu(VXjS) -> ß(Vx,s) as
u —y oo for every /i-continuity set VXjs- Since pu(M \Md) = p(M \Md) =
0, the //-continuity sets of V form a convergence determining class,
i.e. pu A- p on #(R \{0}). Let F denote the distribution function of |X|
and, for n > 1, let an = F"1^ - 1/n) = inf{s G R+ : F(s) > 1 - 1/n}.
Then, as n —y oo, nP(|X| > an) —>• 1 and
"F'!;'x''' = *.. (•) a mo on ß(sd\{o}),nP(|X| > an)
i.e. nP(a"1X G • ) -^ ju(-) on #(Rd\{0}).(i) => (ii) For ar > 0 and S G B^'1),
nP(|X| > anz,X/|X| G S) = nP(a"1X G K,s)
if //(öV^s) = 0. Since, by Theorem 1.14, p has the scaling property
p(xB) = x~ap(B) it follows by Theorem 1.8 that p(dVX:s) = p(Vx,os)
1.3. Multivariate regular variation 23
for every x > 0 and S G B(Sd 1). Hence, with an = (yu(V1)gd-i))1/a;an,
nP(|X| >anz,X/|X| G • ) ^y x~aa(-) onß^"1),
where cr(-) = p(Vi,.)/p(Vi^d-i) is a probability measure on 23(§d_1).For w > ai, let n = n(n) be the largest integer with an < u. For any
x > 0 and S G #(§d_1) we have
nP(|X| >qn+ix,X/|X| G 5) P(|X| >us,X/|X| £ S)
nP(|X| >an)-
P(|X| > u)
nP(|X| >qn£,X/|X| G S)
nP(|X|>an+1)
Moreover, if a(dS) = 0, then
nP(|X|>an+1a;,X/|X|G5)=
n (n + 1) Pff-^X G K,s)
nP(|X|>an)~
n + 1 nF(a^X G 7i^-i)
as u —y oo. Similarly for the upper bound. Hence (ii) holds. D
An immediate consequence of Theorem 1.15 is the following result (seethe proof of Theorem 1.15 for details).
Corollary 1.20 Let \-\a and \ • \b be two norms on Md and let X be
an Md -valued random vector. Then X is regularly varying with index a
with respect to the norm \ • \a if and only z/X is regularly varying with
index a with respect to the norm \ • \b-
It is clear that the corresponding spectral measures do not coincide for
different norms. See Chapter 5 for explicit examples and interpretations
of how the choice of norm affects the spectral measure.
Let us now consider another formulation of multivariate regular vari¬
ation which we will use to show that a random vector with regularly
varying components need not be regularly varying.
Theorem 1.21 Let X be an Md-valued random vector. Then the fol¬
lowing statements are equivalent.
24 Chapter 1. Multivariate regular variation
—d
(i) There exist a relatively compact set E G B(M \{0}) and a nonzero
Radon measure p on B(M \{0}) such that, as u —y oo;
P(XGW-)^M-) onB(Md\{0}). (1.3)(X G uE)
(ii) There exist an a > 0 and a probability measure a on B(Sd x) such
that, for every x > 0, as u —> oo,
x|>^,x/|x|g_o^_m>) onB(Sd-i}<P(|X| > u)
Remark 1.22 If (i) holds, then by essentially the same arguments as
in the proof of Theorem 1.14 there exists an a > 0 such that p(xB) =
x~ap(B) for every x > 0 and B G #(Rd\{0}) and (ii) holds with the
same a. If (ii) holds, then (i) holds and p satisfies the scaling property
above with the same a (see the proof of Theorem 1.15 for details).
Remark 1.23 Note that if (1.3) holds, then
p(xe«,:)=P(X6«-)P(xeug)AifH onB^X{0})P(X e »£) P(X e uE) p(x 6 uE) ß{E)
~ —d ~ ~
for any relatively compact E G B(M \{0}) with p(dE) = 0 and p(E) >
0.
Proof, (ii) =^> (i) Follows by the same arguments as in the proof of
Theorem 1.15.
(i) => (ii) The limit measure p has the scaling property described in
Theorem 1.14 (this can be shown by essentially the same arguments as
in the proof of Theorem 1.14). Hence, by Theorem 1.8, p(dVi^d-i) = 0
and p(dVU:s) = p(VUjos) for every u > 0 and S G B(Sd~1). Hence, if
MK,ös) = 0, then
X| > ux, X/|X| G S)_
P(X G uVXis) P(X G uE)
X| > u) P(X G uE) P(X G nViiSd-i)
^KV*,s)
=x-aKVi,s)
M(^i,sd-i) v(Vi^d-i)
1.3. Multivariate regular variation 25
as u —y oo; i.e. (ii) holds with a(-) = p(V\,.)/p(Vi^d-i). D
The following example shows that a random vector with regularly vary¬
ing components need not be regularly varying.
Example 1.24 Let Q be a probability measure on [0, l]2 such that for
every integer n > 1, Q assigns mass 2~n, uniformly distributed, to the
line segment between (1 - 2~n, 1 - 2~n+1) and (1 - 2~n+1,1 - 2~n).Define the distribution function C by C(u\, U2) — Q([0, u\\ x [0, U2]) for
ui,U2 G [0,1]. Note that C(wi,0) = 0 = C(0,U2), C(ui,l) = u\ and
C(1,W2) = U21 i.e. C is a so-called copula (see e.g. Joe [25]). Note also
that for every n > 1 (with C(u, u) = 1 — 2u + C(u, u))
C(l - 2~n+\ 1 - 2-n+1)/2~n+1 = 1
and
Ü(l - 3/2n+1,1 - 3/2n+1)/(3/2n+1) = 2/3.
In particular lim^i C(u,u)/(1 — u) does not exist. Take a > 0 and let
F be given by F(x) = 1 — x~a for x > 1. Finally, let X be an R2-valued
random vector with distribution function given by ¥(X^ < x\, X^1' <
X2) = C(F(xi),F(x2))- Since X^ and X^ have distribution function
F, it follows by Corollary 1.18 that they are regularly varying. However,
P(XeF-i(«)((l,oo)xR))-6("',')/(l «), (1-4)
which as we have seen does not have a limit as u 11 •If X were regularly
varying with some limit measure p, then by Theorem 1.8 (1, 00) x (1, 00)and (1, 00) x R would be relatively compact //-continuity sets of positive
//-measure and thus, by Theorem 1.21 and Remark 1.23, the left-hand
side of (1.4) would converge as u t 1- We conclude that X is not
regularly varying.
Let us now consider yet another possible formulation of multivariate
regular variation; there exists an a > 0 and a slowly varying function
L (L is a strictly positive Lebesgue measureable function on (0, 00)
26 Chapter 1. Multivariate regular variation
satisfying lim^^oo L(ux)/L(u) = 1 for every x > 0) such that for every
x G Rd\{0}
,.P«x,X) >u) . .
. n . n .
hm —--— = wix.) exists and is finite (1.5)
u^oo U~aL(u)W V ^
and w(x) = 0 is possible for some but not all x G Rd\{0}. It follows
immediately that the function w is homogeneous, w(ux.) = waw;(x) for
every u > 0 and x G Rd\{0}. It is easy to show that (1.2) implies
(1.5). In Basrak, Davis and Mikosch [4] the following theorem proves
equivalence of (1.2) and (1.5) under some additional assumptions.
Theorem 1.25 LetX. be an Md -valued random vector. Then (1.2) and
(1.5) are equivalent if either (i) a is a positive noninteger or (ii) X has
nonnegative components and a is an odd positive integer.
Let X be a random vector satisfying (1.2), i.e. X is regularly varying.
Then X satisfies (1.5) and the limit function w is uniquely determined
by a and a. On the other hand, the limit function w determines a but
not necessarily the spectral measure a if a. > 0 is a positive integer.
Consider the following example.
Example 1.26 Fix an integer a > 1. We will construct two regularly
varying random vectors Xi and X2 with tail index a and different
spectral measures P((cosOi, sinOi) G • ) and P((cos02, sin82) G • ),such that the limit functions w\ and u>2 in (1-5) coincide. Let Oi be
a [0, 27r)-valued random variable with density function fi(9) > w for
9 G [0, 2tt) and some w > 0. Take v G (0, w) and let O2 have density
function /2 given by
hiß) = fi(0) + vsin((a + 2)9), 9 G [0, 2tt).
Let R ~ Pareto(a), i.e. ¥(R > x) = x~a for x > 1, be independent of
Qi, i = 1, 2, and put
A/ RcosGi \
'~ I RsinGi J'
1.3. Multivariate regular variation 27
Take x G R2\{0} and let ß G [0, 2tt) be given by
x_
( cosß
Ixl I sin/5
Then, for u > |x|
x,X2> >u)-P((x,Xi) >u)
= P((x/|x|, X2) > u/|x|) - P«x/|x|, Xi) > u/|x|)/»oo /»/3+arccos((u/|x|)/r)
= v / ar-a-1sin((o; + 2)^)d^drJu/\x\ Jß—arccos((u/|x|)/r)
/»oo />/3+arccos((u/|x|)/r)
= v ar'-1 sin((a; + 2)0))d0drJu/\x\ Jß—arccos((u/|x|)/r)
/»oova r _a_x
'u x
r"a_i cos((a + 2)(ß + arccos((w/|x|)/r)))
— cos((a + 2)(ß — arccos((u/|x|)/r))) jdr/»OO
sin((a + 2)ß) / r~a_1 sin((a + 2) arccos((u/|x|)/r))drAt/lxl
2t>Q!
OL -\- L./u/lx
Using standard variable substitutions and trigonometric formulas the
integral can be rewritten as follows:
/»OO
/ T»-«-1 sin((a + 2) arccos((w/|x|)/r))drAt/|x|
ra_1 sin((a; + 2) arccos(r))dr-a I
i———/ cosa_1(r) sin((a + 2)r) sin(r)dr
lxla Jo
u-a rir/2'
cosa_1(r) cos((a + l)r)dr0
/»TT/2cosa(r) cos((q! + 2)r)dr.
x-a
u-ar/2
-a I
JO
The two last integrals are zero for every a G (0, oo); these integrals can
be found in Gradshteyn and Ryzhik [19] p. 392. Hence, for u > |x|,
P((x, X2) >u)= P((x, Xi> >u). m
28 Chapter 1. Multivariate regular variation
Theorem 1.25 (i) is proved in Basrak, Davis and Mikosch [4] by show¬
ing that the limit function w in (1.5) uniquely determines the spectral
measure a in (1.2) if a > 0 is not an integer. The above example shows
that the idea behind the proof can not be extended to integer-valued
tail indices. Whether the result still holds in this case is not known.
1.4 Sums of regularly varying random vec¬
tors
In this section we will derive some useful results concerning sums of
regularly varying random vectors. The results generalize known results
in the univariate case to the multivariate setting but the techniques used
in the proofs are quite different. Let us start with a result on linear
transformations of a regularly varying random vector (see also Basrak,
Davis and Mikosch [3]). Note that the assumption on X in Theorems
1.27 and 1.28 is weaker than that of multivariate regular variation since
we do not require that the limit measure is nonzero.
Theorem 1.27 Let X be an Md-valued random vector and suppose
there exist a sequence (an), 0 < an | oo, and a Radon measure p
on ß(Rd\{0}) with /i(Rd\Rd) = 0 such that nP(a~lX e ) A p(-) on
B(M^\{0}). If T : Md -+ MP, p < d, is a linear transformation of full
rank, then
nF(a~1T(X) G •) ^y poT'^ DMP) on B(M?\{0}). (1.6)
Proof. Let B e B(m7\{0}) be relatively compact with p o T~1(dB n
MP) = 0. If B C RP\RP, then
n¥(a~1T(X) £ B) = p o T~X(B n Mp) = 0
for all n. Hence we can without loss of generality assume that B D
MP =£ 0. Since B is relatively compact there exists an e > 0 such that
infxGßnKp |x| > e. For x G B fl Rp take y such that T(y) = x (since
1.4. Sums of regularly varying vectors 29
T is onto such a y exists). Since |T(y)| < ||T|||y| and ||T|| < oo for
all linear transformations T : Md -> MP, |y| > |x|/||T|| > e/||T|| for
all y G T~l(B n Rp). Hence T~l(B n MP) G #(Rd\{0}) is relatively
compact. Since T is continuous and p(M \Md) = 0,
0 = p(T~l(dB n Rp)) = p(T~l(d(B n Mp) n MP)) = p(dT~l(B n MP))
and hence
nF(a~lT(X) G B) = nF(a~1T(X) G B n Rp)
= nP(T(a"1X) £5nlp)
= nP(a"1XGT-1(SnRp))
->• //(t-^sdrp))
as n —y oo, from which the conclusion follows. D
We proceed by considering sums of a fixed number of independent reg¬
ularly varying random vectors.
Theorem 1.28 Let X be an Md-valued random vector and suppose that
there exist a sequence (an), 0 < an t oo, and a Radon measure p on
B(M^\{0}) with p(Md\Md) = 0 such that nP(a"1X G • ) ^ p(-) on
£(rA{0}).
(i) If X is a random vector in Md, independent of X, and if there
exists a Radon measure p on B(M \{0}) with p(M \Md) = 0 such
that nP(a"1X G • ) A- p(-) on #(Rd\{0}); then
nF(a~1(X + X) G • ) -A p(-) + p(-) on B(M^\{0}).
(ii) If for some k, there exist independent and identically distributed
random vectors Xi,..., X& such that X = Xi + • • • + X&, then
t
nP(a"1Xi G • ) A -p(-) on B(m\{0}).K
Remark 1.29 A statement similar to (i), for Revalued random vec¬
tors, is proved in Resnick [33] (Proposition 4.1 p. 85).
30 Chapter 1. Multivariate regular variation
Proof, (i) Take t\ > 0 and 2 > 0 and note that, by Theorems 1.8 and
1.14, p(d(Bc0jJ) = p(d(B^J) = 0. Since
nF(a-1(X,X)eBc0^ixBc0je2)= nF(a~1X G Bgjei) P(a"1X G Bg>ea) - 0,
as n ->• 00, it follows that nP(a"1(X,X) G • ) -A p(-) on #(R2d\{0}),where p is a Radon measure which concentrates on ({0} x Md) U (Md x
{0}). Let T : R2d - Rd be the linear transformation T(x, x) = x + x.
By Theorem 1.27,
nP(a"1(X + X) G •) ^/2°^_1(-nRd) on #(Rd\{0}).
Hence for any B G #(Rd\{0});
poT~l(BnMd) = £({(x,x) :x + ÏG£nRd})= £({(x,0) :x + 0G5nRd})
+£({(0,x) :0 + ÏGSnRd})= /z(S)+£(S).
(ii) Since nF(a~1X G • ) A //(•) on #(Rd\{0}) it follows that for any
subsequence (nj) such that nj —y 00 as j —y 00, n^ P(a~1X G • ) A p(-)
on #(R \{0}). Hence, it follows by (i) that any subsequential vague
limit p\ of (nF(a~lXi G • )) must satisfy p\ = p/k. Hence, we only
need to show that {nF(a~1X\ G • )} is relatively compact in the vague
topology. By Theorem 1.4, {nF(a~1Xi G • )} is relatively compact in
the vague topology if and only if supn>1 nF(a~1Xi G B) < 00 for every
relatively compact B G B(M \{0}). We prove this by contradiction.
Suppose that there exists a relatively compact set B G B(M \{0}) such
that supn>1nF(a~1Xi G B) = 00. Then there exists an r > 0 such
that supn>1 nF(a~1Xi G Bq r) = 00. Since nF(a~1Xi G Bq r) < 00
for every n this implies that limsupn^.00 nF(a~lXi G Bq r) = 00. Take
1.4. Sums of regularly varying vectors 31
e G (0,r/k). Then
P(a"1X G Bc0jr_ke)= F(a-1(Xl + ... + Xk)eBc0,r_ke)> F(a~1X1 G Bc^r, a~lX3 G B0,e for j = 2,..., fc)
= P(a"1X1 G ^jr)P(a"1X1 G ßo.e)*-1.
Hence
limsupnP(a~1X G 5o,r-fce)
> limsupnP(a-1Xi G 5£;T.) P^Xi G ßo^)^"1 = oo.
n—>-oo
This contradicts the assumption that {nP(a~1X G • )} is relatively
compact and we conclude that {nF(a~1Xi G • )} must be relatively
compact in the vague topology. D
Above we considered a sum of a fixed number of terms. Let us now
consider the case with a random number of terms N, where N is inde¬
pendent of the terms Xk.
Theorem 1.30 Let (Xk)k>i be a sequence of independent and identi¬
cally distributed Md -valued random vectors and suppose that there exist
a sequence (an), 0 < an t oo, and a nonzero Radon measure p on
B(Md\{0}) with p(Md\Md) = 0 such that nF(a~lX1 G • ) A p(-) on
B(M \{0}). If N is a nonnegative random variable with X^Lo^C-^ =
n)(l + e)n < oo for some e > 0 and N is independent of (Xk)k>i, then
N
nF(a-1J2xk -)AE(N)p(-) onB(Md\{0}).k=i
If (Yk)k>i is another sequence of independent and identically distributed
Md-valued random vectors, independent of N, such that nF(a~1Yi G
• ) A- 0 as n —y oo, then nF(a~l Y2k=i ^k £ ' ) A 0 as n ^ oo.
Proof. Take a relatively compact B G #(Rd\{0}) with p(dB) = 0.
Since for all n, nF(a~1 J2k=i Xfc e M.d\Md) = E(N)p(M.d\Md) = 0, we
32 Chapter 1. Multivariate regular variation
may without loss of generality assume that B cMd. Let 7 = infx_e |x|.Since
N
^XfcGB)n
k=l
oo
= ^nP(a-1^X,Gß)P(AT = 0
00 £
< ^nPfc1 ]T |Xfc| > 7) F(AT = 0 (1.7)1=1 k=i
and |Xfe| is univariate regularly varying, it follows from Theorem 3
in Embrechts, Goldie and Veraverbeke [15] that the right-hand side of
(1.7) converges to ~E(N)p,(Bg ). Hence, using Pratt's Theorem (seePratt [31]) and Theorem 1.28, we conclude that we may interchange
the sum and the limit to obtain
N oo I
lim n F(a~1 V Xk G B) = lim V n P«1 V Xfc G B) F(N = I)n—>-oo ^-—' n—>-oo *-—' ^-—'
fc = l / = 1 fc = l
00 I
= V lim nF(a-1J2xkeB)F(N = l)
= E(N)p(B)
n—>-oo
Z=l fc=l
for every relatively compact B G ß(R \{0}) with p(dB) = 0. The
second claim is proved similarly. D
Remark 1.31 The moment condition on N, J2=o^(N = n)(l + e)n <
00 for some e > 0, comes from the remark following Theorem 3 in
Embrechts, Goldie and Veraverbeke [15]. This result concerns random
sums of univariate random variables with subexponential tails. Since
we use this result for terms which are regularly varying the moment
condition can in fact be substantially weakened.
The following lemma can be used for instance in connection with The¬
orem 1.28 to show that if we add any random vector with all moments
finite to an independent regularly varying random vector X with limit
1.4. Sums of regularly varying vectors 33
measure p, then the sum is regularly varying with the same limit mea¬
sure p.
Lemma 1.32 Let X be an Md-valued random vector. J/E(|X|m) < oo
for every m G N, then for every regularly varying sequence (an), 0 <
an t oo, and every relatively compact set B G B(M \{0}), nP(a~1X G
B) —y 0 as n —y oo.
Proof. Fix a > 0 and let (an), 0 < an î oo, be a regularly varying
sequence with index 1/a. Let B G B(M \{0}) be relatively compact.
Then there exists an x > 0 such that B C Bq x. Hence, for n large,
nP(a^X G B) < nF(a~1X G 5£J = nP(|X| > anx).
Define / by f(t) = inf{n G N : an > t}. Then by Theorem 1.5.12 p. 28
in Bingham, Goldie and Teugels [7] / is regularly varying with index a,
i.e. f(t) = taL(t) for some slowly varying function L, and f(an) ~nas
n —y oo. Hence
nP(|X| > anx) ~ /(an)P(|X| > anx) as n —y oo.
By Markov's inequality we have P(|X| > tx) < E(|X|m)/(te)m < oo for
every t > 0, x > 0 and m > 0. Hence, for every ra > 0,
/(an)P(|X| > anx) < aZL(an)a-mx-mE(\X\m) = C(m)al-mL(an).
Taking m > a and letting n —>• oo yields nP(|X| > ana:) -^ 0 from
which the conclusion follows. D
An immediate consequence is the following result.
Corollary 1.33 Let X be an Md-valued regularly varying random vector
with tail index a > 0 and spectral measure a with respect to the norm
| • |, and let b G Md be a constant vector. Then X+ b is regularly varying
with the same tail index and the same spectral measure with respect to
the norm I • I.
Chapter 2
Regular variation for
multivariate additive
processes
Stochastic processes with heavy-tailed marginal distributions have be¬
come increasingly important in many applications such as communica¬
tion networks, hydrology, insurance mathematics and mathematical fi¬
nance. Many interesting examples can be found in the collections Adler,
Feldman and Taqqu [1] and Rachev [32]. A lot of effort has been put
into studying such processes and into finding the tail asymptotics of the
distribution of functionals of the processes (see e.g. Embrechts, Goldie
and Veraverbeke [15], Rosihski and Samorodnitsky [36] and Braverman,
Mikosch and Samorodnitsky [9]). However, whereas problems concern¬
ing the tail behavior of univariate stochastic processes have been studied
successfully for quite some time, multivariate processes have received
far less attention. In this chapter we focus on multivariate additive
processes, i.e. stochastically continuous processes with independent in¬
crements, which at some fixed time t > 0 are regularly varying, and
we study the tail behavior of vectors of functionals acting on such pro-
35
36 Chapter 2. Regular variation for additive processes
cesses. By tail behavior we mean a regular variation limit measure. The
basic intuition underlying all the results is the following: the process
reaches a set far away from the origin by making one big jump at some
time t < t, and in comparison to the size of the jump, the process does
not move much before r nor between r and t.
We prove tail equivalence between the distribution of Xt and its associ¬
ated Levy measure vt in the sense that if pt is a nonzero Radon measure
on #(Rd\{0}) with ^(Rd\Rd) = 0, then
nP(a"1X, G • ) A pt(-) on £(Rd\{0}) (2.1)
if and only if
niyt(an-) A pt(-) on B(m\{0}). (2.2)
This is a multivariate version, in the regularly varying case, of a result
in Embrechts, Goldie and Veraverbeke [15] which says that
F(Xt > x) ~ vt({y e R : y > x}) as x - oo,
for Xt subexponential. Moreover, we determine the implications of
regular variation of Xt on the joint tail behavior of vectors of functionals
acting on the underlying process (Xs)s>q. For example, we study the
componentwise suprema
X,* = (sup Xi1),..., sup X^)0<s<t 0<s<t
and the componentwise suprema of the jumps
XiA = (sup AX,..., sup AX^),0<s<t 0<s<t
and we show that if the additive process (Xs)s>0 satisfies the regular
variation condition (2.1), then
nF(a~1X*t e-)A pt(-) and nP(a"1Xf £-)A pt(-)
on #(R+\{0}), i.e. we have convergence to the same limit measure. In
this case, knowing the joint tail behavior of X^ is enough to determine
the joint tail behavior of X£ and XA respectively. However, there are
37
many other interesting choices of functionals for which the joint tail
behavior cannot be determined by that of Xt alone. Typically, we would
need information of the type: the probability that Xs for some s G [0, t]reaches a set anBs, where Bs is allowed to vary with s. This can be
d
formulated in terms of a vague limit on B([0, t] x (R \{0})) for the graph
{(s, Xs) : 0 < s < t} of our process. We prove tail equivalence between
the graph of the process and the associated measure v, where v([0, s] x
B) = vs(B) for s G [0, t] and B G #(Rd\{0}), in the sense of having the
same vague limit on #([0,£] x (R \{0})). Since the precise formulation
of the result (Theorem 2.11) requires some additional notation and a
few technicalities, we refrain from stating it at this point. Using this
result we can determine the joint tail behavior of the componentwise
integrals
•I X^ds,...o
in the sense of the vague limit of nF(an1It G • ) on B(M \{0}). As a
special case, if (Xs)s>o is a Levy process, then it follows that
nP«1!, e-)A^-p(-) onB(Md\{0}),a + V
where p and a > 0 are such that Pt(-) = tp,(-) and p(u •) = u ap(-) for
u>0.
The chapter is organized as follows. We begin in Section 2.1 by intro¬
ducing additive processes and recall some of their properties. In Section
2.2 we prove the main results on tail equivalence and joint tail behavior
of functionals acting on the components of the processes.
The symbol ~ denotes both asymptotic equivalence, i.e. f(x) ~ g(x)as x —y oo if linx^oo f(x)/g(x) = 1, and that a random vector has
a certain probability distribution, i.e. X ~ F means that X has the
distribution F. The dual use of ~ should not cause any confusion.
38 Chapter 2. Regular variation for additive processes
2.1 Additive processes
In this section we introduce additive processes following Sato [38]. An
additive process (Xt)t>o on Md is a stochastically continuous stochastic
process with independent increments, starting at zero. There exists a
version of it which has right-continuous sample paths with left limits.
We will always choose such a version. If in addition (Xt)t>o has station¬
ary increments, then it is called a Levy process. For an additive process
(Xt)t>o on Md, for every t, the distribution of X^ is infinitely divisible,
i.e. for any positive integer n there exist independent and identically dis¬
tributed random vectors Zi;ri)i,..., ZnjT1)i such that Xt = Y^i-i ^i,n,t-If (Xt)t>0 is a Levy process, then we can take Zijn>i = Xti/n-Xf(i_1)/n.
We denote by F the characteristic function of a probability distribution
F onRd,
F(z) = / e*<z'x)F(dx).
For any infinitely divisible probability distribution F on Md we have the
Lévy-Khintchine representation (Sato [38] Theorem 8.1 p. 37):
F(z) = exp ( - i<z, Az) + z<7,z> (2.3)
+ / (eiM ~ 1 - t<z,x)l{x:|x|<1}(x))i/(dx)Y</R<*\{0}
J
z G Md, where A is a symmetric nonnegative definite d x d matrix, v is
a measure on Rd\{0} satisfying
/ (|x|2Al)i/(dx) < oo, (2.4)jRd\{0}
and 7 G Md. We call (A, v,~{) the generating triplet of F. The matrix
A and the measure v are called, respectively, the Gaussian covariance
matrix and the Levy measure of F. When A = 0, F is called purely
non-Gaussian.
An important result for additive processes is the Lévy-Itô decomposition
2.1. Additive processes 39
which we will recall below. First some notation. Let
D0)6 = {x G Rd : a < |x| < b}, for 0 < a < b < oo,
Da,oo = {x G Rd : a < |x| < oo}, for 0 < a < oo.
Theorem 2.1 (Sato [38] Theorem 19.2 p. 120) Let (Xt)t>0 be an
additive process on Md defined on a probability space (Q, T, F) with sys¬
tem of generating triplets ((At, vt,~ft))t>o and define the measure v
on (0,oo) x (Rd\{0}) by u((0,t] x B) = ut(B) for B G #(Rd\{0}).Let Qq G T, P(Oq) = 1, be such that t \-y Xt(u;) is right-continuous
in t > 0 with left limits in t > 0 for each u G Q0 and define, for
H eB((0,oo) x (Rd\{0}));
\ 0, foruj^üo-
Then the following hold.
(i) {£,(H) : H G B((0, oo) x (Rd\{0}))} is a Poisson random measure
on (0,oo) x (Rd\{0}) with intensity measure v.
(ii) There exists Q\ G T with F{£l\) = 1 such that, for any to G Q\,
XKlü) = lim / {x£(d(5,x),ü;) -xï7(d(s,x))} (2.6)
+ / xf(d(s,x),u;)J(0,t]xDltOO
is defined for all t G [0, oo) and the convergence is uniform in t
on any bounded interval. The process (X^) is an additive process
on Md with ((0,z^,0)) as the system of generating triplets.
(iii) Define
X?(u;) = Xt(uj) - X](uj) for ooeüi.
There exists Q2 £ J7 with P(f22) = 1 such that, for any u G f22,
X\(lS) is continuous in t. The process (Xf) is an additive process
on Md with ((^,0,7^)) as the system of generating triplets.
(iv) The two processes (Xj) and (Xf) are independent.
40 Chapter 2. Regular variation for additive processes
In connection to this theorem we give the supplementary result which
says that the part of the additive process containing the small jumps
will always have finite moments of all orders. This will be relevant when
studying the tails of a regularly varying additive process.
Lemma 2.2 Let Yt = lime^o J(0t]xD {x£(d(s,x)) —xi/(d(s,x))}. Then
for every t > 0 and m G N, E(|Yf |m) < oo.
Proof. Fix arbitrary t > 0 and integer m > 1 (the case m = 0 is
trivial). For notational convenience, let Zj, Xj and Ytj denote the jth
component of z, x and Y^ respectively. Since any two norms on Md
are equivalent we may without loss of generality take | • | to be the
standard Euclidean norm, i.e. the norm given by |x| = (52i=1x2)1'2.Note that by the Lévy-Khintchine representation for infinitely divisible
distributions
j (ei<z,x> _ x _ ^x^^dx) and f |x|V(dx)
exist finitely. Let Yf = f(0t-\xD (x£(d(s,x)) — xî/(d(s,x))}. Then
Yf A- Yt as e \. 0 and hence the characteristic functions converge,
E(e*<z'Y*>) = exp{ j (e*<z'x) - 1 - i(z,x»i/t(dx)}
-* exp{ j (ei(z'x)-l-z(z,x))^(dx)} =E(e^z'Yt)).
If n\,..., nd G N with rij > 1 for some j G {1,..., d}, then
/ x2ni •
...
• x2dndvt(dx) < f xf-3vt(àx) < j |x|2^(dx) < oo.
Jdoa Jd0i1 Jd0i1
Moreover,
E(|Y<I2"*) = J2 „,"*'„ ^Y*T rM").ft ]_ • • • • 't'c/ •
where since nj > 1 for some j,
= f x\ni •...•x2dndvt(d*) <oo.
Jd0>1
2.2. Regular variation of functionals 41
Hence E(|Yi|2m) < oo. However, E(|Yi|m) < (E(\Yt\2m))1/2 from
which the conclusion follows. D
2.2 Regular variation for multivariate ad¬
ditive processes and for vectors of func¬
tionals acting on such processes
We now turn to the main topic of this chapter; the tail behavior of
multivariate additive processes and functionals of them. In the uni¬
variate case it is well known (see e.g. Embrechts, Goldie and Veraver¬
beke [15]) that for an infinitely divisible regularly varying (or even
subexponential) random variable X with Levy measure v it holds that
F(X > u) ~ v({x : x > u}) as u —y oo. This property is sometimes
referred to as tail equivalence of X and v. For a univariate additive
process (Xt)t>o with system of generating triplets ((At, Vf>lt))t>o this
implies that if Xt is regularly varying (subexponential) for some t > 0,
then F(Xt > u) ~ Vt({x '• x > u}) as u —y oo. The intuition behind
this result is explained by what is sometimes referred to as the "large
deviations principle" : unlikely events happen in the most likely way. In
this case this is interpreted as follows. The most likely way the process
becomes large at time t is due to one big jump of the process before
time t. The Levy measure of {x : x > u] gives the intensity of jumps to
this set during (0, t] and the probability of exactly one jump to this set
is asymptotically ut({x : x > u}). The same intuition holds also in the
multivariate case as the next result shows. The tail equivalence should
now be interpreted as equality of the limiting measures associated with
multivariate regular variation.
Theorem 2.3 Let (Xt)t>o be an additive process on Md with system of
generating triplets ((At, ^ti1t))t>o- Fix an arbitrary t > 0. Then the
following statements are equivalent.
(i) There exist a sequence (an), 0 < an f oo, and a nonzero Radon
42 Chapter 2. Regular variation for additive processes
measure pt on B(M \{0}) with pt(M \Md) = 0 such that
nF(a~1Xt G • ) A pt(-) on B(M~f\{0}). (2.7)
fzz) There exist a sequence (an), 0 < an f oo, and a nonzero Radon
measure pt on B(M \{0}) wz£/& pt(M \Md) = 0 swc/i that
nut(an-) ^y pt(-) on B(Md\{0}). (2.8)
Furthermore, the sequences (an) and the measures pt in (i) and (ii) can
be taken to be equal.
Remark 2.4 (i) Note that for any t > 0 and any infinitely divisible
random vector Y there exists an additive process (Xs)s>0 such that
Y = Xt. Hence Theorem 2.3 can be reformulated in terms of infinitely
divisible random vectors.
(ii) Note also that if Theorem 1.25 (i) was true for arbitrary tails indices
a > 0, then Theorem 2.3 would be easily proved using the results in
Embrechts, Goldie and Veraverbeke [15] for univariate subexponential
infinitely divisible random variables.
Proof of Theorem 2.3. To prove this theorem we will make use
of the Lévy-Itô decomposition, Theorem 2.1, which says that X^ has
representation
Xt=Yt+Jt + X2 a.s,
where Yt, Jt and X2 are independent and, with the notation of Theorem
2.1,
Yt = lim/ {x£(d(S,x))-x£(d(S,x))},
Jt = [ x£(d(s,x)),J(0,t]xDl!OO
Xt = Xt — Yt — Jt •
X2 is Gaussian and hence has finite moments of all orders. By Lemma
2.2, Yt has finite moments of all orders and will therefore, as we will
2.2. Regular variation of functionals 43
see, not contribute to Xt being regularly varying. It will be sufficient
to consider the part Jt being the accumulated big jumps of the process
up to time t. Since £ is a Poisson random measure, the characteristic
function of Jt is
E(ei(z'J*>) = exp { / (e*<z'x) - l)^(dx))
and hence Jt has representation as a compound Poisson random vector
Nt
Jt = / ,Jk,t,k=0
where Nt ~ ~Po(vt(Dij00)), i.e. Nt is Poisson distributed with parameter
vt(DijOQ), J0,t = 0, Jk,t ~ vt( • n Dii00)/vt(Dii00) and all vectors are
independent.
(i) =>• (ii) To prove this implication we will show that
nF(a-1Jhte-)Apt(-)/ut(D^00) as n ^ oo. (2.9)
Then the implication follows since Jk,t ~ vt( • n.Di)00)/i^(.Di)00) and for
any relatively compact B G B(M \{0}) and large enough n, vt({anB} D
-Di,oo) = ut(anB). So if (2.9) holds, then also
nut(an- ) A- pt(-) as n —y oo.
The proof goes in two steps. First we show that for (2.7) to hold it is
necessary that
nF(a~1Jt G • ) A pt(-) as n ^ oo, (2.10)
and for (2.10) to hold it is necessary that (2.9) holds. We start by show¬
ing that the set {nF(a~1Jt G • )} is relatively compact in the vague
topology. As in the proof of Theorem 1.28 we prove this by contradic¬
tion. Assume that there exists a relatively compact B G B(M \{0}) such
that supn>1 nF(a~1Jt G B) = oo. Then there exists an r > 0 such that
B C Bq rand hence supn>1nF(a~1Jt G Bq r) = oo. Since nF(a~1Jt G
Bqt) is finite for every n this implies limsupn_>.00 nF(a~1Jt G Bq r) =
44 Chapter 2. Regular variation for additive processes
oo. Take e G (0,r/3). Then
lim sup nF(a~1Xt G #o,r-2e)n—>-oo
= lim sup nP^"1^ + Jt + Xf2) G ßg>r_2e)n—>-oo
> lim sup nP«1J, G 50c,r)HWYt G A),e)P«%2 e ßo,e)n—>-oo
= OO.
Thus {nF(a~1Xt G • )} is not relatively compact which is a contradic¬
tion. We conclude that {nF(a~1Jt G • )} is relatively compact. Let
(ni) be a subsequence such that lim^oo rii = oo and let pij be the
vague limit of (n(F(a~1Jt G • )) as i —)• oo. Since Yt and X2 have fi¬
nite moments of all orders, by Lemma 1.32, for every relatively compact
BeB(Md\{0}),
ni P(a~1Yi eß)40 and m F(a~^X2 G B) -> 0 as i -+ oo.
Then, by Theorem 1.28 (i), niP(a~1Xt G • ) A //M(-) as i ^ oo.
However, we have assumed that (2.7) holds so p,\j = Pt- Hence, (2.10)holds.
We continue with a similar argument to show that (2.9) holds. Let us
start by showing that the set {nF(a~lJ\j G • )} is relatively compact in
the vague topology. Assume that there exists a relatively compact B G
d
B(M \{0}) such that supn>1 nF(a~ J\,t G -ß) = oo. Then there exists
an r > 0 such that B C B^r and hence supn>1 n P(a~1Ji,t G Bq r) = oo.
Since nF(a~1Jijt G Bq r) is finite for every n, limsup^.^ nF(a~1Jiit G
Bq r) = oo. We have
limsupnP^Ji G B^r) > limsupnP(a-1Ji,t G B^r)F(Nt = 1) = oo.
Thus {nF(a~lJt G • )} is not relatively compact which is a contradic¬
tion. We conclude that {nF(a~lJ\j G • )} is relatively compact. Let
(nj) be a subsequence such that lim^oo nj = oo and let p2,t be the
vague limit of (ujF(a~lJ\j G • )) as j —y oo. By Theorem 1.30 it
follows that
n, PK/J* e ) A E(Nt)p2,t(-) = vt(Di,oo)p2,t(-) as j ^ oo
2.2. Regular variation of functionals 45
and hence we must have p2,t = Pt/^t(Di^oo)- Hence, (2.9) holds.
(ii) => (i) Since Jk,t ~ vt( • D Dij00)/vt(Dij00) and we have assumed
that nvt(an- ) A- pt(-) as n —y oo, it follows that
nP(a~1Jfc;i G • ) A- pt(-)/vt(Di,oo) as n ^ oo
since for any relatively compact B G B(M \{0}) and large enough n,
fi({an.B} n -Di,oo) = vt(anB). Then, by Theorem 1.30,
nPK1^ G • ) ^ E(^)M*(-)M(^i,oo) = Mt(-) as n ^ oo.
Now, since Xt = Y* + J^ + X2 a.s. where the terms are independent, Y*
and X^ have finite moments of all orders and nF(a~1Jt G • ) -A pt(-) as
n —y oo, the conclusion follows by combining Lemma 1.32 and the first
part of Theorem 1.28. D
In this chapter we focus on additive processes. However, in order to
avoid having to prove essentially the same result twice (also in the fol¬
lowing chapter) we formulate Theorem 2.7 below for strong Markov pro¬
cesses. We first show that additive processes are indeed strong Markov
processes (see Remark 2.6 below for details on what is meant by strong
Markov process).
Theorem 2.5 An additive process (Xt)t>o on Md is a strong Markov
process.
Proof. By Remark 1 following Theorem 7 p. 61 in Gihman and Sko-
rohod [21] (and as seen from the proof of Theorem 7) a Markov process
(Yt)t>o on Md is strong Markov if it has a right-continuous version and
if for any bounded continuous / and t > 0 it holds that F defined by
F(s,x) = Es^(f(Yt)) is continuous at (s,x) for any (s,x) G [0,t) x Md.
For an additive process (Xt)t>0 on Md we have (see e.g. Sato [38] p. 56
and 57)
F(S,x) = E*'x(/(Xf))
= J /(y)i^(x, dy) = J /(x + yK*(dy),
46 Chapter 2. Regular variation for additive processes
where p.sj denotes the distribution of Xt — Xs. Fix arbitrary x G Md,
s,t G R+ with s < t and a bounded continuous /. Consider sequences
(sn) and (xn) such that sn —y s and xn —y x. Define, for y G Md and
Bend,
fn(y) -/(x„ + y), /oo(y) = /(x + y),
pn(B) = pSn,t(B), Poo(B) = ps,t(B),
and note that fn —> /oo pointwise and /in -^ //qo- Let E be the set of
y such that fn(yn) —> Zoo(y) fails to hold for some sequence (yn) con¬
verging to y. By Theorem 5.5 p. 34 in Billingsley [5], pn -A p^ implies
Mn ° /n_1 ^ Moo ° /oo1 if Poo(E) = 0. Since fn converges to /^ uni¬
formly on compact sets and since /oo is continuous, E is empty. Hence
the hypothesis of Theorem 5.5 is satisfied. Introduce random variables
Xn and Xoo such that Xn ~ pn o f-1 and X^ ^^0 /-1. Since /
is bounded, {fn} is uniformly bounded and hence {Xn} is uniformly
integrable. Finally, by Theorem 5.4 p. 32 in Billingsley [5] and upon
transformation of integrals,
J /„(z)/zn(dz) = J zpnof-1(dz)=E(Xn)
-y E(XOQ) = / zpoo of~1(dz) = / /oo(z)^oo(dz).
Hence, F is continuous at (s,x) for any (s,x) G [0,t) xMd. D
Remark 2.6 By strong Markov process we mean a Markov process
which satisfies Definition 2 p. 56 in Gihman and Skorohod [21]. In
particular, a strong Markov process is not necessarily temporally ho¬
mogeneous. Note that essentially only one property of strong Markov
processes is used in this and the following chapter, namely that
E(l{r<i}l{xt-xTe^})=E(l{r<t}E^x^(l{Xt-xTeßoV}))for t > 0, r > 0 and a hitting time r.
The next result (a generalized multivariate version of the result in
Willekens [40]) will be of relevance for the subsequent study of func¬
tionals of additive processes, but is also interesting in itself. It says es¬
sentially the following. If the tails of the process are sufficiently heavy,
2.2. Regular variation of functionals 47
then the probability that the process reaches a set far away from the
origin before time t > 0 is asymptotically equal to the probability that
the process ends up in that set at time t. Note that condition (2.11) is
much weaker than that of multivariate regular variation.
For positive r, u and T let
arjT(u) = sup{PSjt(x, B^r) :xGRd and s,t E [0,T],t- s G [0,u]}.
Theorem 2.7 Let (Xt)t>o be a strong Markov process on R and let
A G Vß be bounded away from 0. For a sequence (an) satisfying 0 <
an f oo let rn = infjs : Xs G anA}. Fix t > 0, let r > 0 be arbitrary
and put Sr(anA) = {x G Md : infyGanA |x — y| < r}. Then
(i) F(Xt G Sr(anA)) > F(rn <t)(l- ar,t(t))
(ii) If arjt (t) —y 0 as r —> oo and for any r > 0
P(X g Sr(ar,A))=
(2u)
n^oo p(Xt G anA)' v '
then
lim J^L- = i.n^oo P(Xi G anA)
Proof. Write t = rn. Clearly,
P(r < t) = F(t <t,Xte Sr(zA)) +P(r < t,Xt G Sr(zA)c)
< F(Xt G Sr(zA)) + P(r <t,Xte Sr(zA)c)
< F(Xt G Sr(zA)) + P(r < t, Xt - XT G ßg>r).
By the strong Markov property,
F(r<t,Xt-XTeBc0jr) = E(l{T<nl{Xt_xTGBoV})= E(l{T<nE^x^(l{Xt_xTGi?oV}))< E(l{T</}ar,t(t))= F(T<t)arit(t).
48 Chapter 2. Regular variation for additive processes
Hence, P(Xt G Sr(zA)) > P(r <t)(l- ar:t(t)). Furthermore,
, . r•
tnrn<t)
^vF(rn<t)
1 < hm ml —— jt-< hm sup
n^oo P(Xf G anA)~
n^oo F(Xt G anA)
F(Xt G Sr(anA)) 1< lim sup
n^oo P(Xt G anA) 1 - ar,t(t)1
1 - arjt(t)'
Since r > 0 was arbitrary we can let r —)• oo from which the conclusion
follows. D
Before we proceed we note that for additive processes the condition
®r,t(t) —» 0 as r —)• oo always holds.
Lemma 2.8 Let (Xi)i>0 be an additive process on Mr. Then, for every
t > 0, arj(t) —y 0 as r —>• oo.
Proof. By Theorem 10.4 p. 55 in Sato [38], (Xf)t>0 is a Markov process
with spatially homogeneous transition function
PU;V(x, B) = F(XV - Xu G B - x), 0 < u < v,
where ß-x = {y-x:yG B}. Hence
aTit{t) = sup{PUjV(x, B^r) :xG Md,u,v G [0,t],u< v}
= sup{PUjV (0, BcQr) : u, v G [0, t], u < v}
= sup{P(Xv - Xu G BcQr) :u,ve[0,t],u<v}
< F( sup |Xr -Xu\>r)^0u,v£[0,t]
as r —y oo. D
Theorem 2.9 Le£ (Xi)i>0 èe a separable strong Markov process on
Fix an arbitrary t > 0 and suppose that ar^(t) -> 0 as r —>• oo. XTien
£/ie following statements are equivalent.
2.2. Regular variation of functionals 49
(i) There exist a sequence (an), 0 < an f oo, and a nonzero Radon
measure pt on B(M \{0}) with pt(M \Md) = 0 such that
nF(a~1Xt e-)A pt(-) on B(Md\{0}).
(ii) There exist a sequence (an), 0 < an f oo, and a nonzero Radon
measure pt on B(M \{0}) with pt(M \Md) = 0 such that
nPfû^X, G • some s < t) A pt(-) on B(Md\{0}).
The sequences (an) and the measures pt in (i) and (ii) can be taken to
be equal.
Proof, (i) =>• (ii) Let Vu,s = {x G Rd\{0} : |x| > u,x/|x| G S} for
u > 0 and let S G #(8d_1) be a /it-continuity set. Suppose first that
ßtiYu,s) = 0. For every 8 G (0,1) there exists nrj such that for n > nrj$
Sr(anVu,s) C an(l - ô)VUjs U an(l - S)Vu,Ss(s)\s,
where Sr{anVUjs) = {x G Rd : MyeanVuS |x - y| < r} and SÔ(S) =
{x G Sd~1 : infy<Es |x — y| < ô}. Furthermore, VUjss(s)\s fails to be a
^-continuity set for at most countably many 6. Hence, by Theorem 2.7
(i), for n > nrjS,
nF(a~1Xs G VU:s some s < t)
<nF(a-1Xt G (l-ô)Vu,s)
+nP(a"1Xt G (1 - <5)KÄ(s)\s)
^0 +
1 — «r,« (*) l-"r,<W
(i-(y)-a^(K,5J(5)\g)1 -ar7t(t)
for some a > 0. Since <5 G (0,1) was arbitrary we can let ô —y 0 and the
conclusion follows.
Now, suppose pt(VUjs) > 0. We first show that the condition of The¬
orem 2.7 (ii) is satisfied. Take r > 0. For every ô G (0,1) there exists
nrjs such that for n > nTj$
Sr(anVu,s) C an(l - ô)VUjs U an(l - 5)VUjSg(s)\s-
50 Chapter 2. Regular variation for additive processes
Hence
1 <P(Xt G Sr(anVu,s))
F(Xt G anVu,s)
<P(Xt G aw(l - ô)Vu,s) P(Xt G flw(l - £)K^(s)\s)
P(X* G anVu,s) F(Xt G anK,s)
_^pt((l - ô)VU:s)
|Mt((l-^)K,gg(g)\g)
MK,s) /J*(K,s)
Vt(VUiss(s)\s)\= (l-S)-a 1 +
Mt(K,s)
for some ck > 0 as n —>• oo. Since ô G (0,1) was arbitrary, by letting
<5 —^ 0 it follows that the condition in Theorem 2.7 is satisfied. Hence,
by Theorem 2.7,
P(Xg G anVu,s some s <t)_
n^So P(Xt G anVUts)
and thus
nP(Xs G anVUjs some s < t)
wy r- \r ^vX* G awK,g some g <t)=
nF(Xt G anVu,s) F,Y c „ t/ ^y M*(K,s),
as n —> oo. Since convergence of every such set VUjs implies vague
convergence on B(M \{0}) the conclusion follows.
(ii) => (i) For any A G ß(Rd\{0}) we have F(a~1Xt G A) < F(a~1Xs G
A some s <t) and hence
limsupnP(a^1Xt G A) < pt(A).n—)-oo
A lower bound is constructed as follows. Let VUjs = {x G Rd\{0} :
|x| > u, x/|X| G S} for u > 0 and 5 of the form {x G §d_1 : |x - x0| <
ro} for some xq G Sd~1 and ro > 0, such that VU}s is a /^-continuity
set. For such an S and small enough ô > 0, let S6(S) = {x G 5 :
infyGSd-i\s |x - y| > 6} and for r > 0 let Sr(anVU:S) = {x G a„K,s :
infyGanyc |x — y| > r}. For every ô G (0,1) there exists nrj such that
for n > nrj,
Sr(anVUjS) D a„(l + £)K,s«(s)-
2.2. Regular variation of functionals 51
Hence, for n > nr^,
nF(a~1Xt e Vu,s)
> nF(a~1Xs G (1 + ô)VU}S6(S) some s <t,
Xq - Xs G anB07s all q G [s, t])
> nF(a~1Xs G (l + S)VUisHs) some s < t)(l - a^/^) \
-^(l + ô)-apt(Vu,sS(s)),
by combining the strong Markov property, the assumption arj(t) —y 0
as r —y oo and Lemma 2 p. 420 in Gihman and Skorohod [20]. Since
ô > 0 was arbitrary we can let ô —y 0 and hence
liminfnP(a-1Xf G Vu,s) > Pt(Vu,s).
Since convergence of every such set VU}s implies vague convergence on
ß(Rd\{0}) the conclusion follows. D
We now turn our attention to vectors of functionals applied to each
component of a multivariate additive process. We consider the implica¬
tions of regular variation of the process at time t on the vector of the
componentwise suprema of the process and the componentwise suprema
of its jumps up to time t. We have the following result.
Theorem 2.10 Let (Xt)t>o be an additive process on Md with system
of generating triplets ((At, i^,7f))t>o. Fix an arbitrary t > 0. Then the
following statements are equivalent.
(i) There exist a sequence (an), 0 < an f oo, and a nonzero Radon
measure pt on B(M \{0}) with pt(M \Md) = 0 such that
nF(a~1Xt e-)A pt(-) on B(Md\{0}).
(ii) There exist a sequence (an), 0 < an f oo, and a nonzero Radon
measure pt on B(M \{0}) with pt(M \Md) = 0 such that
nPfû^X, G • some s < t) A pt(-) on B(Md\{0}),
52 Chapter 2. Regular variation for additive processes
The sequences (an) and the measures pt in (i) and (ii) can be taken to
be equal. Moreover, if any of the statements (i) or (ii) hold, then
(iii) nP(a"%A G • ) A pt(-) on B(Md+\{0}),where XA = (sup0<s<i AXs ,..., sup0<s<i AJQ *) and
(iv) nF(a~1X; G • ) A pt(-) on B(Md+\{0}),where X* = (sup0<s<f Xs(1),..., sup^^ X(sd)).
Proof. The equivalence (i) <^> (ii) follows from Theorem 2.9.
(i) => (iii) Since (Xs)s>0 is an additive process there exists Q0 G T
with P(^o) = 1 such that, for every u G Qo-, Xs(u;) is right-continuous
in s > 0 and has left limits in s > 0. Hence XA > 0 for every wgOq
and s > 0. Let £ be the Poisson random measure given by (2.5). Then,
for any xGl}\{0},
P(Xf < x) = P(£((0,i] x [-oo,x)c) = 0) = exp(-^([-oo,x)c)).
Hence
P(XA < x) = { exP(-^([-°°>x)C)) fOT x e M+\{°}510 otherwise.
By Theorem 1.8, pt(d[—oo,x)c) = 0 for every x G R+\{0} (boundariesof spheres centered at 0 have zero /zt-measure). Hence, by Theorem 2.3,
for every x G R+\{0}
nP(a-%A G[-oo,x)c)
= n(l - exp(-z/t(an[-oo,x)c)))
= n^(an[-oo,x)c)(l + 0(^(an[-oo,x)c)))
^/it([-oo,x)c).
Take a, b G R^_\{0} with a < b. Then
nP(an-%AG[a,b))
= E (-^)il+-+id+1nF(a-1X^ G {[-oc,xltl) x ... x [-oo,^J}c),ü,-..)*d{l,2}
2.2. Regular variation of functionals 53
where Xj\ = a^ and Xj2 = b^ for every j G {1,..., d}. Hence
nPK^XfG^b))^^^)).
Since convergence of every such set [a, b) implies vague convergence on
#(R+\{0}) the conclusion follows.
(i) => (iv) Take x G R+\{0}. Note that if we would only take x G
(0, oo)d then we would not end up with a convergence determining class.
Since X^ > Xt a.s. and pt(d[x., oo)) = 0 (boundaries of spheres centered
at 0 have zero /it-measure)
nPK^X* G [x,oo)) > nF(a~1Xt G [x, oo)) ^ ^([x, oo)),
i.e. liminfj^oo nP(a~1X|c G [x, oo)) > ^([x, oo)). To complete the
proof it remains to show that limsupn_).00 nP(a~1X^ G [x, oo)) <
Pt([x, oo)). First, define
A(1_e)x = {zeMd:z^>(l-e)x^,j = l,...,d}
C%} = {z G Md : z G [-oo, x)c, z^ G [-oo, (1 - e)x^)}
Dgl = {z G Md : z G [-oo, x)c n Ac(1_e)x1 z G [x(fc), oo]}
and note that for each k G {1,..., d}, [—oo, x)c C A(i_e)x U C^J U Dxjand that the sets on the right-hand side are disjoint. If X^ G Ax,
then either Xs G A(!_e)x for some s < t or, for some A; G {1,..., d],
XSl G Cx,e for some s\ < t and XS2 G -Dx,i for some s2 <t,S2=/z s\.
Assume without loss of generality that
P(Xai G C^x,e,XS2 G Da%e some Sl,s2 < t)
> F(XS1 G CikXe, XS2 G £><£>x>e «orne Sl,s2< t)
for fc = 2,..., d. Then
nP(X* Gan[x,oo))
< nP(Xs G Aan(1_e)x some s<t)
+nP(ug=1{Xfll G C£Xj£>X8a G Df)X]£ some 5l,s2 < t})
< nP(Xs G Aan(i_e)x some s < t)
+ndF(XSl G C£>x>eîX82 G L»^ some 5l)52 < t)
54 Chapter 2. Regular variation for additive processes
By Theorem 2.7
nP(Xs G Aan(i_e)x some s<t)
~nF(Xt GAan(1_e)x)-^ Mt((l - e)[x, oo)) = (1 - e)"a/ii([x, oo)),
as n —y oo. Let 7 = inf cr,(i) cn(i) |u — v|. Then 7 > 0 and
ndP(XSl G C£Xf£,Xfl2 G Z^ some 5l,s2 < t)
< ndP(XSl G Ci;)X]£,XS2 G D^ some 5l < s2 < t)
+ ndF(XSl G C^, XS2 G Da%je some s2 < 5l < t)
< ndF(XSl G Ci;)X]£,XS2 - XS1 G 5J>an7/2 some 8l < s2 < t)
+ ndF(XSl G D^x>e, XS2 - XS1 G ßS>0n7/2 some 8l < s2 < t)
< ndF(Xs G C<£Xte some s < t) F(XS G Bc0^ß some s < t)
+ n<2P(Xs G 2#Jx>e some s < t) F(XS G ßS,an7/4 some * < *)
—>• 0 as n —> 00.
Since e > 0 was arbitrary it follows that limsupn_).00nP(a~1X£ G
[x, 00)) < pt([x., 00)). Hence
nF(a~1X; G [x,oo)) ^/if([x,oo)).
Take a, b G R+\{0} with a < b. Then
nP(a-1X;G[a>b))
= J2 (-l)il+-+idnF(a-lXl G [xiil5oo) x ••• x fed,oo)),H,...,id6{l,2}
where £ji = a^^ and #j2 = &(" for every j G {1,..., d}. Hence
fiP(a;1Xt*G[a)b))^/it([a,b)).
Since convergence of every such set [a, b) implies vague convergence on
#(R^_\{0}) the conclusion follows. D
Assertion (ii) of the Theorem 2.10 gives the asymptotics for the prob¬
ability that the process reaches a certain set during the time inter¬
val [0,t]. What if we would allow the set of interest to vary over
2.2. Regular variation of functionals 55
time? I.e. we are looking for the asymptotics of the probability that
the graph of the process, {(s,Xs) : 0 < s < t], intersects a rela¬
tively compact set in #([0,£] x (R \{0})), the cr-algebra generated by
the sets of the form T x B with T G B([0,t]) and B G #(1^(0}).This requires knowledge of the tail of Xs for (almost) all s G [0,t],not only of Xt as has been the case so far. To obtain this kind of
result we will work with the measure v (see Theorem 2.1) instead of
simply ut. In order to be able to use the vague convergence frame¬
work we extend the measure v to [0, oo) x (R \{0}) by requiring that
u({(s,x) : s = 0 or x G R \Rd}) = 0. This extension is unique. Let us
introduce the operation * such that for a G (0, oo) and sets of the form
A x B, A G B([0,t]) and B G #(Rd\{0}), we have a * A x B = AxaB.
Clearly this operation can be extended to all sets in B([0, i]xl \{0}).
Theorem 2.11 Let (Xt)t>o be an additive process on Md with system
of generating triplets ((At, vt,^t))t>o- Fix an arbitrary t > 0. Then the
following statements are equivalent.
(i) There exist T C [0,t] such that 0,t G T and such that [0,t]\T is
at most countable, a sequence (an), 0 < an f oo, and a collection
of Radon measures {ps : s £T} on B(M \{0}) such that for every
s G T, ps(M \Md) = 0, pt is nonzero, and
nF(a~1Xs G • ) A ps(-) on B(m\{0}). (2.12)
(ii) There exist T C [0,t] such that 0,t G T and such that [0,t]\T is
at most countable, a sequence (an), 0 < an f oo, and a collection
of Radon measures {ps : s G T} on B(M \{0}) such that for every
s G T, ps(M \Md) = 0, pt is nonzero, and
nvs(an-)^ps(-) onB(Md\{0}). (2.13)
(iii) There exist a sequence (an), 0 < an f oo, and a nonzero Radon
measure p on B([0,t] x (Rd\{0})) with p([0,t] x Rd\Rd) = 0 such
that
nV(an * • ) A p(-) on B([0, t] x (Rd\{0})). (2.14)
56 Chapter 2. Regular variation for additive processes
(iv) There exist a sequence (an), 0 < an f oo, and a nonzero Radon
measure p on B([0,t] x (Rd\{0})) with p([0,t] x Rd\Rd) = 0 such
that
nP({(«, Xs) : 0 < s < t} n (an * • ) ^ 0) A /!(•) (2.15)
on B([0,t]x (Rd\{0})).
Furthermore, the sequences (an) in (i)-(iv) can be taken to be equal and
then the measures ps and p([0, s]x ) in (i)-(iv) coincide on B(M \{0})
for every s G [0,i\.
Remark 2.12 As seen from the proof below, (iv) =>• (i) with T = [0,t]and (iii) =>• (ii) with T = [0,t]. Hence we can without loss of generality
let T = [0,t].
Definition 2.13 A stochastic process on R satisfying Theorem 2.11
(iv) is said to have a regularly varying graph on [0,t].
Note that v({s] x B) = 0 for every s G [0,t] and B G #^^{0}).However, the limit measure p may charge sets of the form {s} x B as
seen in the following example.
Example 2.14 Let v be a probability measure on [0,1] x [1, oo) given
by
oo
v(dt x dx) =^2kl^i/2-i/(2k),i/2+i/(2k))(t)dtl[kik+i)(x)ax~a~1dx.fc=i
Let £ be a Poisson random measure with intensity measure v, and let
X = (Xt)tç\o,\\ be a stochastic process given by
Xt = / x£(ds x da:).J[0,t]x[l,oo)
2.2. Regular variation of functionals 57
For any t G (0,1),
ù([t - l/(2m),t + l/(2m)] x [1, oo))
< £([1/2 - l/(2m), 1/2 + l/(2m)] x [1, oo))oo
= J2 (k~a -(k+1)_a) = m~a -* °
k=m
as m —y oo. Similarly, v([0, l/m] x [1, oo)) —y 0 as m —y oo and v([l —
l/m, 1] x [1, oo)) —y 0 as m —y oo. Hence, for s < t, v([s, t] x [1, oo)) —)• 0
as \t — s\ —y 0. Thus, for any e > 0,
F(\Xt-Xs\>e) < F(£([s,t]x[l,oo))>0)
= 1 - exp{-v([s,t] x [l,oo))| -> 0
as |t — s| —y 0, i.e. X is stochastically continuous. Moreover, by Propo¬
sition 19.5 p. 123 in Sato [38], for disjoint T1:... ,Tk G B([0,1]),
/ x£(ds x da;),..., / xÇ(ds x da:)JTix[l,oo) JTfex[l,oo)
are independent, i.e. X has independent increment. Moreover, by con¬
struction, X(uS) is right-continuous with left limits for every u G Q,.
Finally, Xq = 0 a.s. Hence, X is an additive process. By Proposi¬
tion 19.5 p. 123 in Sato [38], for every t G [0,1], Xt has Levy measure
ut(-) = £([0,t] x • ). Take u > 0. For t > 1/2 and n large enough,
/>oo
rw([0, t] x (n1//aw, oo)) = n ax~a~1dx = u~a.Jn1/au
For t = 1/2,
1 1°°nù([0,t] x (n1/au, oo)) = n- j
For t < 1/2 and n large enough,
n£([0,t] x (n1/au,oo)) = 0.
Hence, for every t G [0,1],
nvt(nl'a -)^pt(-) on£((0,oo]),
Iax~a~1dx = -u~a.
58 Chapter 2. Regular variation for additive processes
where
T 0 if t< 1/2,tH(B)=i \ JB ax~a~ldx \ît =1/2,
{ JBax~a-1dx ift> 1/2.
In particular, p({l/2} x B) = ^ JBax~a~1dx which clearly may be
nonzero.
The following Lemma is needed for the proof of Theorem 2.11.
Lemma 2.15 Let (Xt)t>o be a strong Markov process on R .Fix an
arbitrary t > 0 and suppose that cvrj(t) —>• 0 as r —y oo. Then, for any
sequence (an) with 0 < an t °°;
nF(a-1Xt G • )-H>0 onB(Md\{0}) (2.16)
z/ and only if
nF(a~1Xs G • some s<t)^0 on B(Md\{0}). (2.17)
Proof. Take a sequence (an), t > 0 and a relatively compact B G
ßor'uo}).Suppose that (2.17) holds. Since
P(X* G anB) < F(XS G anB some s < t),
(2.16) follows immediately.
Suppose that (2.16) holds and, without loss of generality, that B cMd.
Fix r > 0 and let 7 = infxGs |x|. By Theorem 2.7 (i),
Xs G anß some s <t) < F(XS G ünBg^ some s <t)
<[Xt G fl.(qwflS,7))
l-ttr;f(t)
For every r > 0 there exists an nrj7 such that Sr(anBg^) C an-Bo ,2
for n > nri7. Hence
nF(XteanBc /2)limsupnP(Xs E anB some s <t) < lim sup
'—
n—)-oo n—>-oo -L C^r,i i^J
= 0,
2.2. Regular variation of functionals 59
from which (2.16) follows. D
Proof of Theorem 2.11. (i) <^> (ii) For every s G T for which ps
is nonzero this equivalence has been established in Theorem 2.3. We
need to establish the equivalence also if ps = 0. Since pt is nonzero,
by Remark 1.13, the sequence (an) has to be regularly varying. Hence
equivalence can be proved by exactly the same arguments as in the proofof Theorem 2.3.
(ii) =>- (iii) Define the Radon measure p on #([0,£] x (R \{0})) by
p([0,s] x B) = ps(B) for every s G T and B G BQÜ^O}). Since such
sets of the form [0, s] x B form a 7r-system which generates #([0,£] x
(R \{0})) this uniquely defines p. We first need to show that the set
{nv(an * • )} is relatively compact in the vague topology. For each
bounded set A G #([0,£] x R \{0}) there exists a bounded set B G
#(Rd\{0}) such that A c [0,t] x B. Hence,
supnu(an * A) < supnvt(anB) < oo,n n
from which it follows by Theorem 1.4 that {nu(an * • )} is vaguely
relatively compact. Let p! be a vague limit of the subsequence (n'v(an> *
• )). Fix an arbitrary s G T and relatively compact B G B(M \{0}) with
pt(dB) = 0. Then ps(dB) = p([0, s] x dB) < p([Q, t] x dB) = pt(dB) =
0 and hence
n'v(ani * [0,s] x B) = n'us(an'B)
-+ ps(B) = p([0,s]xB),
i.e. /TQOjS] x B) = /l([0,s] x B). Since the sets of the form [0,s] x B
with s ET and B G 23(R \{0}) with pt(dB) = 0 form a 7r-system which
generates #([0,£] x (R \{0})) the conclusion follows.
(iii) =>• (ii) Put ps(-) = p([0,s] x • ) for every s G [0,t]. Let F G
d
B(M \{0}) be closed and bounded. Then [0, s] xF is closed and bounded
and by Theorem 1.1 and Remark 1.2,
lim supnvs(anF) = lim supnv(an * [0, s] x F)n—)-oo n^-oo
< p([0,s]xF) = ps(F).
60 Chapter 2. Regular variation for additive processes
d
Hence, by Theorem 1.1 and Remark 1.2, nus(an • ) -^ ps(-) on B(M \{0})for every s G [0,t].
(i) =>• (iv) Define the Radon measure p, on #([0,£] x (R \{0})) by
p([0,s] x B) = ps(B) for every s G T and B G #(^{0}). Since such
sets of the form [0, s] x B form a 7r-system which generates #([0,£] x
(R \{0})) this uniquely defines p. We first need to show that the set
{nP({(s,Xs) : s G [0,t]} D (an * • ) ^ 0)} is relatively compact in
the vague topology. For each bounded set A G #([0,£] x R \{0}) there
exists a bounded set B G B(M \{0}) such that A C [0,t] x B. Hence,
by Theorem 2.10,
supnP({(s,Xs) :s£ [0,t]} n (an * A) ^ 0)n
< supnP(a~1Xs G B some s < t) < oo,n
from which it follows by Theorem 1.4 that {nP({(s,Xs) : s G [0,t]} n
(fln * • ) 7^ 0)} is vaguely relatively compact. Let p' be a vague limit
of the subsequence (n'P({(s,Xs) : s G [0,t]} n (an' * • ) ^ 0)). Fix an
arbitrary s G T and relatively compact 5 G #(R \{0}) with pt(dB) =
0. Then, by Theorem 2.10 and Lemma 2.15,
n'F({(s,Xs) : s G [0,t]} n (on* * [0,s] x5)^)
= n' P«/Xu G 5 some u < s) -> //S(S) = /I([0, s] x 5),
i.e. /i'([0,s] x B) = /l([0,s] x S). Since the sets of the form [0,s] x 5
with s eT and ß G B(M \{0}) with pt(dB) = 0 form a 7r-system which
generates #([0,£] x (R \{0})) the conclusion follows.
(iv) => (i) Put ps(-) = p([0,s] x • ) for s G [0,t]. Let F G #(Rd\{0})be closed and bounded. Then [0, s] x F is closed and bounded and by
Theorem 1.1 and Remark 1.2,
limsupnP(a~1Xw G F some u < s)n—>-oo
= limsupnP({(s,Xs) : s G [0,t]}n{an*[0,s] x F) ^ 0)n—>-oo
<M0,s]xF)=|is(F).
Hence, by Theorem 1.1 and Remark 1.2, nP(a~1Xu G • some u < s) A-
Ps(-) on B(M \{0}) for every s G [0,t]. Combining Theorem 2.10 and
2.2. Regular variation of functionals 61
Lemma 2.15 now yields nF(a~1Xs G ) A ps(-) on #(Rd\{0}) for
every s G [0, t]. D
Working on the product space [0, t] x (M \{0}) allows us to consider
further interesting functionals such as the integral of each component
of the process. To derive the tail behavior for this functional we use
the intuitive argument described in the introduction which says that if
the process takes a big jump, then it varies very little before and after
the jump, compared to the size of the jump. Intuition then tells us that
the integral of component j, say, becomes bigger than u if Xg exceeds
u/(t — s) for some s G [0,t]. That is, the integral of X^ becomes
bigger than u if the graph {(s,Xg ) : 0 < s < t} intersects with the
set a[3) = {(s,x) G [0,t] x (R\{0}) : x > u/(t - s)}. This argument is
made precise in the final result of the chapter.
Theorem 2.16 Let (Xt)t>o be an additive process on Md with system of
generating triplets ((At,vtilt))t>o- Fix an arbitrary t > 0 and suppose
there exist a sequence (an), 0 < an t °°? and a nonzero Radon measure
p on B([0,t] x (Rd\{0})) with p([0,t] x Rd\Rd) = 0 such that
nu(an * • ) A p(-) on B([0, t] x (Rd\{0})).
-rrtv(i)A„ rt v(d)// It = (f0 X^ds, ...,j; Xsa)ds), then
nF(a-% e-)A £({(*,x) G [0,*] x (r"\{0}) : x G-!-
• })t — s
on B(M \{0}). In particular, if (Xs)s>0 is a Levy process, then
nF(a-1lt G • ) A ^rMO on B(m\{0}),a + 1
where p(-) = p([0,1] x • ) and a > 0 is such that p(u • ) = u~ap(-) for
all u > 0.
Proof. By the Lévy-Itô decomposition we may write the process (Xs)as the sum of three independent processes. With the notation of Theo-
62 Chapter 2. Regular variation for additive processes
rem 2.1,
x» = xJH
+ lim/ {x£(d(it, x),a;) — xi/(d(it, x))}e±° J(o,s]xDeA
+ / xf(d(u,x),o;),J(0,s]xDliOO
J.,
on Qi, where (X^) has a version with continuous sample paths, (Ys) is
a jump process with small jumps, and (Js) is a jump process with big
jumps. For j = l,...,m, let Vf = Jtj/m - Jt(j-i)/m- For u G fi0nßi,
s as m -^ oo,
and
5^|V^(o;)|-)> 5Z lAJ«MI asm^oo,
J= l u(0,<)
where the sum extends over all (finitely many) u such that |AJu(a;)|is nonzero. Take e G (0,1/2) and let A be a relatively compact /it-
continuity set of the form {x G Rd\{0} : |x| > w,x/|x| G S} for
u > 0 and S G jB(8d_1). Convergence of every such set implies vague—d
convergence on B(M \{0}). To begin with, we study the behavior of
nP(J0 Jsds G anA) as n —y oo. Let 7 = infxfE,4 |x| and set At = {(s, x) :
x G t^A, s G [0, t]} G #([0, t] x (Rd\{0})). We begin by constructing a
2.2. Regular variation of functionals 63
lower bound.
m,
n¥C>Z—Jtj/m eanA)3= 1
>nP(U=1{V G \ flw(l + e)A,^|Vri<ttne7/t})t — ti/m
,
m1
= EF(E lV"l < *nej/t) nF(Vf G flw(l + e)A)
771 771 -.
>P(ElVn <<WA)£«P(V -——a„(l + e)^)j=l 3=1
•"
7711
> F(E lV7l ^ «W*) «P(U7=i{V7 G flw(l + e)A}).r—' J J t — ti rn
3 = 1'
Letting ra —>• oo, we arrive at
nF(J Jsds e anA) > F( ^ |AJ„| < anej/t) nv((l + e)an * At).
We now construct an upper bound.
771,
nF£-J^/-Ga-A)lit
3 = 1
< n¥(Uf=1{VT e t-r/m^1 " e)A' ^ |Vrl"
°nC7/*})J/m
i^3
+ nF(Jti/meB^anei/t,
Jtj/m ~ Jti/m G 50,an67/t SOme « < j < "l)
1
£ — £j/ra
+ «P(J«/mGSoian£7/t>
Jti/m - Jti/m G ß0,ane7/* SOme * < 3 < )
1
t—
tj/m
+ nF(Jtj/m G £o,a„e7/t some 3 < )2-
< nP(Uf=1{V7 G ——^-^(l - e)A})
< nP(Uf=1{V7 G^
,,^an(l - e)A})
64 Chapter 2. Regular variation for additive processes
Letting m —y oo, we arrive at
et
nP( / Jsds G anA)Jo
<rw((l-e)an*At)+nF(Js G £o,ane7/* somes G (0,t])2.
Since limn_),ooP(^uG/0iN |AJW| < ane7/£) = 1 and by the second part
of Theorem 2.7 lim-^ nP(Js G -B^ a ,tsome s G (0, £])2 = 0, we get
liminf nP( / Jsds G ani) > p((l + e) * At),n^°° Jo
limsupnP(/ Jsds G ani) < p((l - e) * At).n—>-oo Jo
Since e > 0 was arbitrary and p(u*- ) = u~ap(-) for all u > 0, it follows
that the lower and upper bound coincide, i.e.
lim nP( / Jsds G anA) = p(At).7WOO Jo
Since as n —)> oo
r 1
nP( / Xjds G an4) < nP(Xj G -anA some s < t) -> 0
Jo ^
and
/** i
nP( / Ysds G anA) < nF(Ys G -anA some s < t) -y 0,jo ^
by Theorem 1.28,
lim nP( / Xsds G anA) = lim nP( / Jsds G anA).n-»-oo J0 n->-oo JQ
Hence
lim nP( / Xsds G anA) = //(At).n^°° Jo
In the case of a Levy process, i.e. if /x([0,s] x • ) = sp(-) for s G [0,i\.then
KAt) = [ P(T^-A)ds = p(A) [ (t - sTds = ^-\p(A).Jo t - s Jo a + !
D
Chapter 3
Regular variation for
general stochastic
processes
In this chapter we consider a different and more general approach to
heavy tail analysis for stochastic processes. We will however make sev¬
eral comparisons with the results derived in Chapter 2. Multivariate
regular variation which was discussed in detail in Chapter 1 provides
a natural way for understanding the tail behavior of heavy-tailed ran¬
dom vectors. The definitions of regular variation for random vectors
can be modified so that the modified formulations make sense for more
general state spaces; similar formulations are possible for stochastic pro¬
cesses with sample paths in D([0, l],IRd), the space of Revalued right-
continuous functions on [0,1] with left limits. These formulations seem
to be well suited for understanding the tail behavior of heavy-tailed
stochastic processes. We will exemplify this in various forms through¬
out the chapter. Recall that a random vector X on Md is said to be
regularly varying if there exist an a > 0 and a probability measure a on
the unit sphere Sd~l = {x G Md : |x| = 1} such that, for every x > 0,
65
66 Chapter 3. Regular variation for stochastic processes
as u —y oo,
P(|X|>us,X/|X|e.),x-„j() „.»(s-i-ix
where B(Sd_1) denotes the Borel cr-algebra on Sd~1 and -^ denotes weak
convergence. The probability measure a is referred to as the spectral
measure of X. It describes in which directions we are likely to find
extreme realizations of X. Similarly, we say that a stochastic process
X = (Xt)iG[oti] with sample paths in Z)([0, l],K.d) is regularly varyingif there exist ana>0 and a probability measure a on £>i([0,1], Md) =
{x G D([0,l],Md) : supfr0 ]_] |xi| = 1} such that, for every x > 0, as
u —y oo,
P( X oo> UX,X/\X\00 G • ) _
, ,
o/T-i /rn il in>d\\
p(|xu>u)>* "(•) onB(Dl([0,l],M)),
where B(D1([0,l],Md)) denotes the Borel cr-algebra on £>i([0, l],Md)and Ixloo = supfGr0 ^ \n.t\- The spectral measure a contains essentially
all relevant information for understanding the extremal behavior of the
process X. For example, it might be of interest to know under which
conditions the extremes of X are due to (at most) one single extreme
jump (we allow also an extreme starting point). This can be formalized
in terms of the support of the spectral measure by showing that the
spectral measure concentrates on step functions, i.e. on the set
{xGJDi([0,l],Rd):x = ylK1],^G[0,l],yG§d-1}.
We show that this is the case for a large class of regularly varying Markov
processes, including all regularly varying additive processes (and hence
also all regularly varying Levy processes).
A natural question is why one would prefer formulating regular variation
on D([0, l],Ed) rather than on, say, (E^)^0'1]. The main reason is that
many interesting mappings from D([0,1], Md) to D([0,1], Md) (or to Mk)are continuous whereas the corresponding mappings from (R^)^0'1] are
not. However, with constructions similar to the one used in this chapter,
regular variation can be formulated on other complete separable metric
spaces. In this chapter we prefer to work on D([0,1], Md). An equivalent
67
definition of regular variation on D([0,1], Md) is the following; a stochas¬
tic process X with sample paths in D([0, l],Kd) is regularly varying if
there exist a sequence (an), 0 < an t oo, and a nonzero boundedly finite
measure m on B(D([0,l],Md)) with m(D([0, l],Rd)\D([0, l],Md)) = 0
such that, as n —y oo,
nP(a"1XG •) Âm(-) onB(D([0,l],Md)), (3.1)
where A denotes so-called ^-convergence. (The precise meaning of
D([0, l],Md) is explained in Section 3.1. At this point it may be viewed
as only a slight modification of D([0,1], IRd) needed in order to use the
concept of Û7-convergence.) Let h be a positively homogeneous (i.e.
fo(Ax) = A/i(x) for A > 0) measurable mapping from D([0, l],IRd) to
D([0,1], M?) (or to Mk). Then, if (3.1) holds and if h satisfies some mild
conditions, as n —y oo,
nF(a~lh(X) G • ) A m o h~1{ • n D([0,1], Md)) on B(D([0,1], Md))(3.2)
k
(or on B(M \{0}), M = [—00,00]), i.e. we have a version of the Con¬
tinuous Mapping Theorem. Hence, under mild conditions on h, regular
variation of X implies regular variation of h(X) and we can express its
limit measure in terms of m and h as in (3.2). In Section 3.1 we state
the two definitions of regular variation on D([0, l],Md) and show that
they are equivalent. Moreover, we give necessary and sufficient condi¬
tions for regular variation for a general stochastic process with sample
paths in D([0, l],IRd). Finally, we give a continuous mapping theorem
which provides a powerful tool in the subsequent analysis. In Section 3.2
we focus on strong Markov processes with asymptotically independent
increments (see Section 3.2 for the precise meaning of asymptotically
independent increments). We obtain sufficient conditions for regular
variation for such processes which are easier to verify since they involve
only the marginals X^ of the process X. Moreover, we show that the
limit measure m of such regularly varying Markov processes vanishes
on Ve where
V = {x G £>([0, l],Ed) : x = yl[W)1],t; G [0, l],y G Md\{0}}.
This means that, asymptotically, the process reaches a set far away from
the origin either by starting there or by making exactly one big jump to
68 Chapter 3. Regular variation for stochastic processes
this set and, in comparison to the size of the jump, it stays essentially
constant before and after the jump. For an illustration, see Figure 3.1
which shows eight simulations of
x | {\xs\ > i^ixV0-9)for some s G I°> X]}'
where X is a Levy process with X\ having a Cauchy distribution with
density fxx(x) = l/(ir(l + x2)). On one hand this means that we are
able to quantify the idea of one big jump in terms of the support of
the regular variation limit measure. On the other hand, and equally
important, this in combination with the Continuous Mapping Theorem
(3.2) allow us to explicitly compute tail probabilities of h(X) for many
interesting choices of h. See e.g. Examples 3.20 and 3.21 with
Mx) = (vte[o,i] *e[o,i]
(1) (d)sup xt ,..., sup xt
and
h(x)=(J1x[1)dt,...,J1xld)dt)respectively. In Section 3.3 we study filtered stochastic processes of the
form
Yt= f f(t,s)dXs, te [0,1], (3.3)Jo
where X is a regularly varying Levy process (i.e. a strong Markov pro¬
cess of the type studied in Section 3.2) with sample paths of finite
variation. Under the assumption that the kernel / is continuous we
show that Y can be viewed as a mapping of the process X, which is
sufficiently regular to satisfy the conditions of the Continuous Mapping
Theorem. We show that Y is regularly varying and determine the limit
measure.
In order to make the presentation as readable as possible and in order to
focus the attention on the underlying ideas rather than on technicalities,
we give the proofs at the end of each section.
3.1. Regular variation on D 69
3.1 Regular variation on D
Let us introduce regular variation on D = D([0, l],Rd); the space of
functions x : [0,1] —y Md which are right-continuous with left limits.
This space is equipped with the so-called Ji-metric (referred to as do in
Billingsley [5]) which makes it complete and separable. The formulation
of regular variation we will use has recently been introduced in de Haan
and Lin [14] in connection with max-infinitely divisible distributions on
D. See also Giné, Hahn and Vatan [18].
We denote by D\ = £>i([0,1], Md) the subspace {x G D : suptGr01i |xt| =
1} equipped with the subspace topology. Define D = (0, oo] x _Dl5 where
(0,oo] is equipped with the metric p(x,y) = \l/x — l/y\ making it
complete and separable. Then D is a complete separable metric space.
Note that to each nonzero function x G D corresponds a unique element
(#*,x) G D where x* = supfGr01-i |x^| and x = x/x*. For x G D we
write Ixloo = suptGr0-n \x.t\ and for x = (x*, x) G D we write jx^ = x*.
A consequence of the above construction is that
B(D)n(D\{0}) = B(D)n(D\{0}),
i.e. the Borel sets we are interested in are the usual Borel sets on D
which do not contain the zero function.
We will see that regular variation on D is naturally expressed in terms
of so-called w-convergence of boundedly finite measures on D, i.e. mea¬
sures which assign finite measure to bounded sets. A sequence of bound¬
edly finite measures (mn)n<=^ on a complete and separable metric space
E converges to m in the tû-topology, mn A- m, umn(B) —y m(B) for ev¬
ery bounded Borel set B with m(dB) = 0. If the state space E is locally
compact, which D is not but M \{0} is, then a boundedly finite mea¬
sure is called a Radon measure, and w)-convergence coincides with vague
convergence and we write mn -^ m. Finally we note that if mn A m
and mn(E) —>• m(E) < oo, then mn A m. For details on w-, vague- and
weak convergence we refer to Appendix 2 in Daley and Vere-Jones [12].See also Kallenberg [26] for details on vague convergence.
70 Chapter 3. Regular variation for stochastic processes
Definition 3.1 A stochastic process X = (Xt)t^[o,i] with sample paths
in D is said to be regularly varying if there exist a sequence (an),0 < an t °°; and a nonzero boundedly finite measure m on B(D) with
m(D\D) = 0 such that, as n —y oo,
nF(a~1X e ) A m(-) onB(D). (3.4)
If Definition 3.1 holds, then the limit measure m has the following scal¬
ing property (the proof is identical to that of Theorem 1.14, with the
obvious notational changes, and therefore left out).
Theorem 3.2 The limit measure m has a scaling property; there exists
an a > 0 such that m(uB) = u~am(B) for every u > 0 and B G B(D).
An equivalent and perhaps more intuitive formulation of regular vari¬
ation on D is given in the next result. Its proof is identical to that of
Theorem 1.15 with obvious notational changes.
Theorem 3.3 Let X = (Xt)t(=[o,i] be a stochastic process with sample
paths in D. Then the following statements are equivalent.
(i) X is regularly varying in the sense of Definition 3.1.
(ii) There exist an a > 0 and a probability measure a on B(Di) such
that, for every x > 0, as u —y oo,
P(|X|00>^,X/|X|00G-)^ _a
p(|xu>u)>* *(•) onB(Dl). (3.5)
Remark 3.4 If (i) holds, then by Theorem 3.2 there exists an a > 0
such that m(uB) = u~am(B) for every u > 0 and B G B(D) and (ii)holds with the same a. If (ii) holds, then (i) holds and m satisfies the
scaling property above with the same a (see the proof of Theorem 1.15
for details).
Definition 3.5 For a stochastic process X satisfying (3.5) we refer to a
and a as the tail index ofX and the spectral measure ofX, respectively.
3.1. Regular variation on D 71
The two formulations of regular variation on D given above are much
inspired by the formulations of regular variation for random vectors.
Many of them are documented in e.g. Basrak [2].
Remark 3.6 For S G B(D{), let VijS = {x G D : |x|oo > l,x/|x|oo G
S}. It follows from the proof of Theorem 3.3 that the probability mea¬
sure a and the boundedly finite measure m are linked through
Let h : D —>• D or h : D —y Mk be a measurable, positively homogeneous
mapping, i.e. h(Xx) = A/i(x) for A > 0 and x G D. If X is a regularly
varying stochastic process with sample paths in D we may be interested
in the tail behavior of h(X). This is achieved using an analogue of the
Continuous Mapping Theorem for weak convergence. Let D^ = {x G
D : h is discontinuous at x}. Note that D^ G B(D) (see Billingsley [5]
p. 225) and hence also Dh n D G B(D).
Theorem 3.7 (Continuous Mapping Theorem) LetX = (Xt)t<=[o,i]be a stochastic process with sample paths in D. Suppose that there exist
a sequence (an), 0 < an t oo, and a nonzero boundedly finite measure
m on B(D) with m(D\D) = 0 such that, as n —y oo,
nP(a"1X G • ) Â m(-) on B(D).
Let h : D —) D be a positively homogeneous measurable mapping such
that /i_1(.B) is bounded in D for every bounded B G B(D) D D and
suppose m(Dh f\D) =0. Then,
nF(a~1h(X) G •) Âmoh~l(- HD) onB(D).
Moreover, the result holds for mappings h : D —y Mk with the obvious
notational changes.
The formulation of regular variation on D in combination with Theo¬
rem 3.7 allow us to derive the tail behavior of a large class of continuous
72 Chapter 3. Regular variation for stochastic processes
mappings of stochastic processes. This will be illustrated in the follow¬
ing sections.
The next theorem gives necessary and sufficient conditions for a stochas¬
tic process with sample paths in D to be regularly varying. Before stat¬
ing these conditions we introduce some notation. For x G D, To C [0,1]and 6 G [0,1] let
w(x,T0) = sup{|xs -xt| : s,t G T0},
w"(x,ô) = sup min{|xt-xtl|,|xt2-xt|}.t1<t<t2,t2-ti<8
Theorem 3.8 Let X = (Xt)t(=[o,i] be a stochastic process with sample
paths in D. Then the following statements are equivalent.
(i) There exist a set T C [0,1] containing 0 and 1 and all but at most
countably many points of [0,1], a sequence (an), 0 < an t °°> and
a collection {mtl...tk ' k G N, ^ E T} of Radon measures with
m(M \Mdk) = 0, and mt nonzero for some t £T, such that
_. dh
nF(a-1(Xt1,...,Xtk)e-)Amt1...tk(-) on B(M \{0}) (3.6)
holds whenever t\,..., tk G T. Moreover, for any e > 0 and r\ > 0,
there exist a ö G (0,1) and an integer no such that
nF(w"(X,Ô) > ane) < m n>n0, (3.7)
nF(w(X,[0,ô))>ane)<r], n>n0, (3.8)
and
nF(w(X, [l-ô, 1)) > ane) <n, n > n0. (3.9)
(ii) There exist a sequence (an), 0 < an t oo; and a nonzero boundedly
finite measure m on B(D) with m(D\D) = 0, such that
nF(a~1Xe -)Âm(-) onB(D). (3.10)
The sequences (an) in (i) and (ii) can be taken to be equal. Moreover, the
measure m in (ii) is uniquely determined by {mtl...tk ' k G N, ti G T}.
3.1. Regular variation on D 73
Most stochastic processes with sample paths in D and regularly varying
finite dimensional distributions that appear in applications are regularly
varying on D. However, in order to fully understand the conditions of
Theorem 3.8 (i) we find it relevant to study examples of stochastic
processes for which one of the conditions (3.7), (3.8) and (3.9) does
not hold. In Example 3.17 below we construct an additive process
which satisfies all conditions of Theorem 3.8 (i) except (3.9). Roughly
speaking, this additive process is constructed so that for arbitrary small
5 > 0 the probability of extreme jumps within the time interval [1 — 6,1)is too high. Consider also the following example which illustrates a
violation of condition (3.7).
Example 3.9 Let a > 0 and consider independent random variables
Z and V where Z ~ Pareto(o;), i.e. F(Z > x) = x~a for x > 1, and V
is uniformly distributed on [0,1]. Let (Yt)t>o be given by
0 ifte[0,V),
Y=,Z ifte\V,V + l/(2Z)),
1 S0 iî te [V + 1/(2Z),V + 1/Z),z if te [v + i/z,œ),
and let X = (Xt)te[o,i] be given by Xt = Yt for t G [0,1]. Then X
satisfies (with an = n1/") all conditions of Theorem 3.8 (i) except (3.7):for any e > 0 and ö G (0,1)
nF(w"(X,ô) > n1/ae) ~nP(Z> n1/ae) = e~a
as n —y oo.
3.1.1 Proofs
Proof of Theorem 3.7. Let Nh = {x e D : h(x.) = 0} and define
7^ : D\Nh ^Dby
Mx) =
h(x) if xeD\Nh,x if x G D\D.
74 Chapter 3. Regular variation for stochastic processes
Then L\ C (Dh n~D)\J (D\D), where D\ denotes the set of points of
D\Nh where h is discontinuous. Take arbitrary bounded B e B(D)with m(h~\dB)) = 0. Since dh~l(B) C h~l(dB) Ui\, we have
m(dh~l(B)) < m(h~1(dB)) + m{L\) = 0. Hence
n F(a~1h(X) G B) = n P(a"1/i(X) G B, h(X) / 0)
= n F(a~lh(X) e B, h(X) / 0)
= nF(a~1X G h~\B) n (D\Nh))
= nF(a~lXeh~1(B))-+ m(h~1(B)).
Hence, by Proposition A2.6.II p. 628 in Daley and Vere-Jones [12],
nF(a~1h(X) G •) Amoh~l(.) on B(D).
However, for every B e B(D), m(h (B)) = m(h~1(B n D)). Hence
nF(a~1h(X) G •) A m o h'1 (-f\ D) on B(D).
The proof for mappings h : D —y Mk is similar. D
Proof of Theorem 3.8. (i) => (ii) Let mn(-) = nF(a~xX e • ). First
we will show that the set {mn} is relatively compact in the w)-topology.
To prove this we will apply Proposition A2.6.IV p. 630 in Daley and
Vere-Jones [12], which says that it is sufficient that the restrictions
{rnn,j} to a sequence of closed spheres S7 f D are relatively compact
in the weak topology. For 7 > 0, let 57 = {x G D : |x|oo > 7}, and for
n > 1, let mn)7(-) = nP(a~1X G • fl S7). We will show that, for every
7 > 0, the family {mn^} is uniformly bounded and that it is relatively
compact in the weak topology.
Take 7 > 0 and ti,...,tk G T with 0 = t\ < • • • < tk = 1 and
ti — U-i < 6, where 6 > 0 is such that nF(w"(X,ö) > anj/2) < r\
for n > no- Then
mnn(D) = nP(|X|oo > a„7)
< n F( max |XtJ > 0^7/2 or w"(X, Ö) > anj/2)l<i<k
< nF( max |Xt,| > anj/2) + nF(w"(X,6) > anl/2)l<i<k
= fn(l)+9n(l)-
3.1. Regular variation on D 75
By (3.6), (fn(l)) converges to some finite limit as n —y oo and hence
the sequence (fn(l)) is bounded. Moreover, gn(j) < n for n > no, and
clearly gn(j) < no for n < no- Hence, supn>1 mn^(D) < oo, i.e. {mn;7}is uniformly bounded.
Since mn(-) = nF(a~lX G • ) < no for n < no and since a probability
measure P(a~1X G • ) on B(D) is tight it follows by Theorem 15.3 p. 125
in Billingsley [5] that (3.7), (3.8) and (3.9) hold for the finitely manyn preceding no by taking Ö small enough. Hence, we may assume that
no = 1. Note that [7, 00] x K\ G B(D) is compact in D if and only if
K\ is compact in D\. For any n > 0, by (3.7), (3.8) and (3.9), we can
choose Ok such that, if
AK1 = {x G £>i : w"(x, ok) < 1/k},
Ak,2 = {x G £>i : w(x, [0,4)) < 1/fc},
^fc,3 = {x G Di : w(x, [1 - «S*., 1)) < 1/k},
then mn)7([7,00] x (D{\Akj)) < (l/3)n/2fc for every j and n. Let
5 = n^=1 n|=1 Afcj. If K1 is the closure of B, then by Theorem 14.4
p. 119 in Billingsley [5], K\ is compact in D\. Moreover, for every n,
mn>7fD\([7,oo] x K{)) < mre>7([7,oo] x (D{\B))00 3
^ Yl Yl m--7([7,00] x (^A^fc,j))k=lj=l
00
< n^2-fc=n.fc=i
Hence, we have shown that {mTOj7} is uniformly bounded and tight. It
follows from Prohorov's Theorem (Theorem A2.4.I p. 619 in Daley and
Vere-Jones [12]) that {mnjl} is relatively compact in the weak topol¬
ogy. Thus, by Proposition A2.6.IV p. 630 in Daley and Vere-Jones [12],
{nP(a~1X G • )} is relatively compact in the «)-topology. We will
now show that any subsequential w)-limit m satisfies m(D\D) = 0.
By (3.7) and the above argument we can choose u\ and ô such that
nF(w"(X, ô) > anu\/2) < n/2 for every n > 1 (i.e. we may take no = 1
in (3.7)). By (3.6) and Theorem 3.7 (for mappings h : Mk —)> [0,oo))there exist a Radon measure v on i3((0, 00]) with ^({00}) = 0 such that
!/„() := nF(a~1 max |Xtfc | G • ) ^ i/(.) on B((0, 00]).\<i<k
76 Chapter 3. Regular variation for stochastic processes
It follows that v has the scaling property described in Theorem 3.2
(the same proof applies with the obvious notational changes). Hence
there exists an a > 0 such that v([x, oo]) = x~av([l, oo]) for every
x > 0. Choose x such that u([x/2,oo]) < n/4. Then there exists n'
such that vn([x/2, oo]) < n/2 for n > n'. Clearly there exists x' such
that vn([x'/2, oo]) < n/2 for n < n'. Hence, with w2 = max(a;,/),
^([^2/2,00]) < n/2 for every n > 1. Hence, with u = max(wi,w2), for
every n > 1,
nPflXloo > anu) < nP(max |XtJ > anu/2)l<i<k
+nF(w"(X,ö) >anu/2)
< n/2 + n/2 = n.
Suppose mn< -^ m. We have just shown that for any r? > 0 there exists
u > 0 such that mn'({x G D : jx^ > u}) < r\ for n' > 1. In particular,
this implies that m„/({x G D : |x|oo > u}) —y 0 uniformly in n' as
u —»• 00. Since Gu = {x G D : (x^ > w} is open and bounded we have
m(Gu) < liminfn'^-oo nin'(Gu) and because of uniform convergence
m(D\D) = lim m(Gu)
< lim liminfmn'(Gu) = liminf lim mn'(Gu) = 0.u—)-oo ro'—>-oo n'—>-oo «—>-oo
Let m and m be two subsequential u)-limits. We will show that m =
m and that 771 is uniquely determined by {mt1...tk ' k G N, £j G T}.Let Tm and T^ consist of those t G [0,1] for which the projection itt
is continuous except at points forming a set of m-measure 0 and fh-
measure 0, respectively. Then, by Theorem 3.7, for rj1?.. . ,rjfc e Tm n
TjnHT,
mo7Tt-?..tfc( • fit*) = mo7rt-U( • nEdfc) = mtl...tk(-) on £(Rd\{0}).
Since Tm, T^ and T each contain all but countably many points of [0,1],the same is true for Tm D T^ D T, in particular Tm D T^ D T is dense in
[0,1]. Moreover, 0,1 G Tm PlT^ PlT. With some minor modifications of
Theorem 14.5 p. 121 in Billingsley [5] one can show that
{7rriltk(H):keN,HeB(Mdk\{0})nMdk,t1,...,tkeTrnnTfhnT}
3.1. Regular variation on D 77
generates B(D) n D. Hence m and m coincides on B(D) n D and since
m(D\D) = m(D\D) = 0 we have m = m.
(ii) =>• (i) Let Tm consist of those t in [0,1] for which the projection 7rt
from D to Md is continuous except at points forming a set of ra-measure
0. The projections 7ro and 7Ti are continuous and hence 0,1 G Tm. For
t e (0,1), 7Tt is continuous if and only if m({x : xf 7^ *-t-}) = 0. By the
same arguments as in Billingsley [5] p. 124 there are at most countably
many t G (0,1] such that m({x : xf 7^ xt-}) > 0. Then, since m is
nonzero and Tm is dense in [0,1], there exists t G Tm such that mt
is nonzero. Moreover, 7Ttl...tk is continuous except at points forming
a set of m-measure 0 if t\,.. .,tk G Tm. Hence, by Theorem 3.7, for
^li • • • i^k G 1m,
nP(a-1(Xtl,...,Xtfc)G.) = nF(a~1X G ^\Ah{ n Rdfc))
-^ °^U('nRd*) on^(Edfc\{0}).
For t1;..., tk G Tm, let m^...^ (•) = m o tt,"1.^ ( • n Mdk).For n > 1, let mn(-) = nP(a~1X G ). By the scaling property of m,
the set 5„ = {x 6 D : |x|oo > u] is an m-continuity set for every u > 0.
Hence, mn(Su) —y m(Su) = w_am(5'i) for every w > 0. Choose w such
that u~am(Si) < n/4. Then there exists n\ such that mn(Su) < n/2for n > n\. By Proposition A2.6.IV p. 630 in Daley and Vere-Jones
[12], for every 0 < 7 < u < 00, {mn( • D {x G -D : |x|oo G [7,^]})} is
relatively compact in the weak topology on D. Since {x G D : |x|oo G
[7, m]} C .D\{0} and on this subspace the subspace topologies (of D
and D) coincide it follows that {mn( n {x G D : |x|oo G [7, w]})} is
relatively compact in the weak topology on D. Hence, by Theorem 15.3
p. 125 in Billingsley [5], for any e > 0 and n > 0 there exist ö G (0,1)and integer n<i such that
nF(w"(X,ö) > ane, (X^ G an[j,u]) < n/2, n > n2,
nF(w(X, [0,(5)) > ane, JX^ G an[j,u]) < n/2, n > n2,
and
nP(w(X, [1- 6,1)) > ane, (X^ G an[7,w]) < n/2, n > n2.
In particular the three inequalities above hold, with n/2 replaced by n
and n2 replaced by no = max(ni,n2), for u = 00 and 7 < e/2 and for
such 7 they coincide with (3.7), (3.8) and (3.9). D
78 Chapter 3. Regular variation for stochastic processes
3.2 Markov processes with asymptotically
independent increments
In this section we will study Markov processes with increments that
are not too strongly dependent in the sense that an extreme jump does
not trigger further jumps or oscillations of the same magnitude with a
nonnegligible probability. We will derive surprisingly concrete results
for such Markov processes (see Theorem 3.12 and 3.18) which will prove
very useful when used in combination with Theorem 3.7 (see e.g. Ex¬
ample 3.20 and 3.21).
Let (Xf)iG[0)i] be a Markov process on Md with transition function
Ps,t(x, B). For r > 0 and 0 < u < T < 1 define
arjT(u) = sup{PSjt(x, B^r) :xG Md and s,t e [0,T],t- s G [0,u]}.
Note that if the random vectors Y and Y are independent and, for
some sequence (an), 0 < an f oo, and Radon measures m and m with
ra(Ed\Rd) = m(Md\Ed) = 0 we have
nF(a~1Y e •) A m(-) and nP^YG-)^^) on £(Rd\{0}),
then, by Theorem 1.28,
nP(a"1(Y + Y) G -)^m(-)+m(-) onB(Md\{0}), (3.11)
i.e. the limit measure of the sum is the sum of the limit measures.
Independence of Y and Y is not necessary for (3.11) to hold. For
Markov processes the much weaker condition o;T.)i(l) —y 0 as r —y oo is
sufficient (with Y and Y representing two nonoverlapping increments)as shown in the following lemma.
Lemma 3.10 Let (Xt)t<=[o,i] be a Markov process on Md such that
Oir,i(l) —> 0 as r —y oo. Fix arbitrary s,t G [0,1] with s < t. Let (an)be a sequence with 0 < an t oo, and let ms, mt and p be Radon mea¬
sures on B(Md\{0}) with ms(Md\Md) = mt(Md\Md) = p(M^\Md) = 0.
3.2. Asymptotically independent increments 79
Consider the following statements.
nF(a~1Xs G • ) A ms(-) on B(m\{0}), (3.12)
nP(a"1Xf G • ) ^ mt(-) on B(Md\{0}), (3.13)
nF(a-1(Xt-Xs)e-) ^ p(-) on B(M.\{0}). (3.14)
// any two of the above three statements hold, then the third also holds
and the limit measures are related through mt = ms + p.
Lemma 3.10 justifies the following choice of terminology.
Definition 3.11 A Markov process (Xt)t^[o,i] on Md is said to have
asymptotically independent increments if av,i(l) —> 0 as r —y oo.
It turns out that for a strong Markov process (see Remark 2.6) with
sample paths in D and asymptotically independent increments we can
obtain sufficient conditions for regular variation on D, which are easier
to verify than the general conditions of Theorem 3.8.
Theorem 3.12 Let X = (Xf)fG[o7i] be a strong Markov process with
sample paths in D such that ar,i(l) —y 0 as r —y oo. Suppose there
exist a set T C [0,1] containing 0 and 1 and all but at most countably
many points of [0,1], a sequence (an), 0 < an t oo, and a collection
{mt : t e T} of Radon measures on B(M.d\{0}), with mt(Md\Md) = 0
and with mi nonzero, such that
nF(a~1Xt e-)^ymt(-) on B(Md\{0}) for every t G T, (3.15)
and such that, for any e > 0 and n > 0 there exists a 6 > 0, ô G T,
1 — ô G T such that
ms(Bc0je)-m0(Bc0je)<r] and m1(Bc^e) - ml.s(Bc0^) < n. (3.16)
Then there exists a nonzero boundedly finite measure m on B(D) with
m(D\D) = 0, such that, as n —y oo,
nP(a"1X G • ) Â m(-) on B(D).
Moreover, the measure m is uniquely determined by {mt : t G T}.
80 Chapter 3. Regular variation for stochastic processes
Remark 3.13 By Theorem 11.1 p. 59 in Sato [38], a sufficient condition
for a Markov process (Xt)t>o on Md to have a version in D is that
limaeT(u) = 0 for any e > 0 and T > 0.u4-0
Theorem 11.1 considers Markov processes with fixed starting points,
however this requirement can be dropped as seen from the proof. Note
that the above condition implies stochastic continuity.
Remark 3.14 We have chosen to formulate the above theorem and
results below for strong Markov processes, since it is convenient to use
the strong Markov property in some of the proofs. We could however,
instead of the strong Markov property, have assumed that we have just
the Markov property and that, for every e > 0, na2e x(l) —y 0 as
n —y oo.
The following result, for additive processes, was given as Theorem 2.11
in Chapter 2. However, the result and the proof apply in our more
general setting.
Theorem 3.15 Let X = (Xt)t<=[o,i] be a strong Markov process with
sample paths in D such that ar;i(l) —y 0 as r —y oo. Then the following
statements are equivalent.
(i) There exist T C [0,1] such that 0,t eT and such that [0,1]\T is
at most countable, a sequence (an), 0 < an t °°? and a collection
of Radon measures {ps : s G T} on B(M \{0}) such that for every
s eT, ps(M \Md) = 0, pt is nonzero, and
nF(a~1Xs e ) A ps(-) on B(m\{0}). (3.17)
(ii) There exist a sequence (an), 0 < an t °°> and a nonzero Radon
measure p on B([0,1] x (Ë*\{0})) with p([0,1] x M^XM*1) = 0 such
that
nP({(s,Xs) :0<s< 1} D (an * • ) / 0) A p(-) (3.18)
onß([0,l]x (Md\{0})).
3.2. Asymptotically independent increments 81
Furthermore, the sequences (an) in (i) and (ii) can be taken to be equal
and then the measures ps and /l([0,s] x • ) coincide on B(M \{0}) for
every s G [0,1].
Remark 3.16 As seen from the proof of Theorem 2.11, if nF(an1Xs G
• ) A- ps(-) on £(Ëd\{0}) for every seT, then nP(a"1Xs G • ) A ps(-)
on #(Ëd\{0}) for every s G [0,1].
The following example shows that one can find an additive process
X = (Xt)t(=[o,i] which has a regularly varying graph (i.e. satisfies the
conditions of Theorem 3.15 (ii)) but which is not regularly varying on
D (i.e. satisfies neither the conditions of Theorem 3.12 nor those of
Theorem 3.8).
Example 3.17 Let v be a probability measure on [0,1] x [1, oo) given
byoo
v(dt x dx) = ^2kl{1_1/k^](t)dtl[kjk+i)(x)ax~a~ldx.k=i
Let £ be a Poisson random measure with intensity measure v and let
X = (Xt)t£[o,i] be a stochastic process given by
Xt = / xt;(ds x dx).J[0,t]x[l,oo)
Then, by the same argument as in Example 2.14, X is an additive
process and, for every t, vt(-) = ^([0,£] x •) is the Levy measure of Xt.
Take u > 0 and note that
/»oo
n£([0,1] x (n1/au, oo)) = n / ax'^dx = u~a,Jn1/au
and for t < 1 and n large enough,
nv([0,t] x (n1/au, oo)) = 0.
Hence, for every t G [0,1],
nut(n^a -)^pt(-) on£((0,oo]),
82 Chapter 3. Regular variation for stochastic processes
where
r o ift<i,
I SBocx-a~ldx ift=l.
Fix e > 0. For every 5 G (0,1) and large enough n,
nF(w(X,[l-5,l)) >n1/ae)
= nP( sup \XS-Xt\ >n1/ae)s,te[i-6,i)
>nP(C((l-(n1/ae)"1,l) x [n1/ae,oo)) > 0)
= n(l - exp{-£((l - (n1/ae)~\ 1) x [n1/ae, oo))})
> \nv((l - (nllae)-\ 1) x [n^e, oo))Li
> \nv((l ~ ([n1/ae] + 1)"\1) x [[n^e] + 1, oo))
1oo
= 0" E (k~a ~ (k + l)~a)fe = [n1/ae] + l
= ln([n^ae] + 1)"° - ±e~a
as n —y oo. Hence, for n < ^e_a there exists no ô G (0,1) such that
nF(w(X, [1 — J,l)) > n1/ae) < n for all large enough n. Hence, by
Theorem 3.8, X is not regularly varying on D!
It turns out that a regularly varying strong Markov process with sample
paths in D and asymptotically independent increments has a very simple
extremal behavior. In this case the process reaches a set far away from
the origin by making at most one jump to that set (it might start there
at time 0 since we allow for a regularly varying starting point) and the
process essentially stays constant before and after the jump. This is
formalized in the next theorem. Let
V = {x G D : x = yl[Vil],v G [0, l],y G Ed\{0}}.
Theorem 3.18 Let X = (Xt)t(=[o,i] be a strong Markov process with
sample paths in D such that arj\(l) —y 0 as r —y oo. Suppose there exist
3.2. Asymptotically independent increments 83
a sequence (an), 0 < an t oo, and a nonzero boundedly finite measure
m on B(D) with m(D\D) = 0 such that, as n —y oo,
nP(a"1X G • ) Â m(-) on B(D).
Then m(Vc) = 0. Moreover, there exist an a > 0 and a probability
measure a on B(D\) such that, for every x > 0, as u —y oo,
P(|XU>n,,X/|XUG_O^_M0on
P(|X|00>w)
«;i£/i cr({x G Di : x = yl^i], v G [0, l],y G §d_1}) = 1. Moreover, on
a({y:eDl : x = yl[ü;1], v G [0,1], y G • })
coincides with the spectral measure ofXi.
For Levy processes we can be even more explicit.
Example 3.19 Let X = (Xt)t<=[o,i] be a Levy process on Md. Suppose
there exist a sequence (an), 0 < an t oo? and a nonzero Radon measure
mi with mi(ld\Md) = 0 such that
nP(a"1Xi G • ) ^ mi(-) on ß(Rd\{0}).
Since X has stationary and independent increments, nP(a~1Xi G ) A
imi(-) on jB(M \{0}) for every t G [0,1]. Hence, combining Theorems
3.3 and 3.12 gives, for every x > 0, as u —y oo,
P(W,>^,x/|x|,6.)^P(|X|oo > U)
where a > 0 is the tail index of Xi and it follows that
a(-) = F({zi[VA](t),te[o,i]}e-),
where Z and V are independent, the distribution of Z is the spectral
measure of Xi and V is uniformly distributed on [0,1]. The random
vector Z is the direction of the big jump and V is the time of the big
jump. See also Figure 3.1.
84 Chapter 3. Regular variation for stochastic processes
The following two examples illustrate the usefulness of Theorem 3.18
in combination with Theorem 3.7. Compare with the rather technical
proofs needed to prove the corresponding results in the special case of
additive processes (Theorems 2.11 and 2.16).
Example 3.20 Let X be a strong Markov process with Xq = 0 sat¬
isfying the conditions in Theorem 3.18 and let h : D —y Md be defined
by
/i(x) = ( sup x\ ,..., sup x\ '\.MG[0,1] *G[0,1] '
Then it is straightforward to show that h satisfies the conditions of
Theorem 3.7. Hence,
nF(a~1h(X) e ) A m o h~\- n Md) on B(m\{0})
and for B e B(m\{0}), with Mf = [0, oo)d,
mo/r1(ßnRd) = m({xEÖ:/i(x)eßnId}nV)= m({x G D : x = yl[Vil]iv G [0,1], y G B n Md+})= m^-^B nR|)nv)= m^-^BDMJl))= mi(ßnKj),
where mi is vague limit of (nP(a~1Xi G )).
Example 3.21 Let X be a strong Markov process with X0 = 0 sat¬
isfying the conditions in Theorem 3.18 and let h : D —y Md be defined
by
Mx) = (jf x^dt,..., / x[d)dt
Then it is straightforward to show that h satisfies the conditions of
Theorem 3.7. Hence,
nF(a~1h(X) e ) -^ m o h~\- n Md) on B(m\{0})
3.2. Asymptotically independent increments 85
and for B e B(m\{0})
mor1(ßni(l)= m({x G D : h(x) G^nl^nV)= m({x G D : x = yl[Vjl]lv G [0,1], y(l - v) G B n Md})
= m({x G £> : xf G B n Md some £ G [0,1]} D V)J. L
= m({x e D :xt e B n Md some £ G [0,1]}).J. L
In particular, if (Xt)t£[o,i] is a Levy process, then the last expression
reduces to (see also Theorem 2.16)
/ mi(—^—BnMd)ds = mi(BnMd) f (1 - s)ads = —mi(5),Jo 1~s Jo a + !
where mi is vague limit of (nP(a~1Xi G ))
3.2.1 Proofs
Proof of Lemma 3.10. Fix a relatively compact B e B(M \{0}).Then there exist r > 0 such that B C Bqt. Note that the scaling
property implies that sets of the form Bq r,r > 0, are always ms-, mt-
and //-continuity sets.
Suppose that (3.12) and (3.13) hold. We first show that {nP(a"1(Xi -
Xs) G • )} is vaguely relatively compact. We have
supnF(a~l(Xt - Xs) e B) < supnF(a~1(Xt - Xs) G Bc0 r)
< supnP(a"1Xs G B^ r/2)+supnF(a~1Xt G ßgr/2) < oo,n>l
'
n>l'
since {nP(a~1Xs G • )} and {nF(a~lXt G • )} are vaguely relatively
compact. Hence {nP(a~1(Xf — Xs) G • )} is vaguely relatively compact.
By essentially the same argument it follows that if (3.12) and (3.14)
hold, then {nP(a~1Xt G • )} is vaguely relatively compact, and if (3.13)and (3.14) hold, then {nP(a~1Xs G • )} is vaguely relatively compact.
86 Chapter 3. Regular variation for stochastic processes
Suppose that (3.12) and (3.13) hold. Let p be a subsequential vague
limit such that
n'F(a-}(Xt-Xs)e-)^p(-).
Fix ei > 0, Ê2 > 0 and a relatively compact B G B(M \{0}) with
ms(dB) = p(dB) = 0. We have
n'F(a^(Xs,Xt-Xs)eBc0jeixBc0j£2)= n'F(a^Xs e Bc0jei)F(a-ï(Xt - Xs) G B^ \ a~^Xs G B^J
v
v'v
v'
-0.
Since ms(M \Md) = p(M \Md) = 0 we may without loss of generality
assume that B n Md / 0. Then,
n'F(u;(XS)Xt-Xs)eßx%2)= n'Pfc1^ G B)(l - F(a~}(Xt - X.) G Sg>£a | a~}Xs G S))
vv
' >
v'
^SCB) <aa ,e2,i(l)->-0
^ras(£).
Clearly,
n'P(a-1(Xs,Xf-Xs) G B0,eixB) < n'F(a-^(Xt-Xs) e B) ^ p(B).
Set 7 = infxGBnEd |x|. Then
n/P(a-/1(Xs,Xt-Xs)G50,ei x B)
= n'F(a-^(Xt-Xs)eB)- n'F(a^Xs e Bc0^)F(a^(Xt -Xs)eB \ a~}Xs G B^J
>n'F(a-^(Xt-Xs)eB)- n'F(a-^Xs e B^ei)F(a-?(Xt - Xs) G B%„ \ a~^Xs G B^J
vv
/Nv
'
->-m*(Bo,ei) <aan/7,i(l)^0
->/*(£)
It follows that n/P(a-/1(XB,Xt-Xa) G • ) A p(-) on #(R2c\{0}), where
/2 is a Radon measure which concentrates on ({0} x Md) U (Md x {0}).
3.2. Asymptotically independent increments 87
Hence
ri F(a~ï(Xs + Xt-Xs)e-)^ /j((x, x) : x + x G • ),
where
//((x,x) :x + xG •) = /2((x,0) :x + 0 G • )+/2((0,x) : 0 + ÏG • )
= ms(-)+p(-).
However, n'F(a~^(Xs-\-Xt — Xs) G • ) -4- mt(-) and hence p = mt—ms.
Since this is true for any subsequential vague limit of (nP(a~1(Xi —
Xs) G • )) it follows that
nP(a"1(Xt - Xfl) G • ) A mt(-) - ms(-).
Suppose now that (3.12) and (3.14) hold. By the same arguments as
above, replacing n' by n, it follows that
nP(a"1XfG-) = nF(a-1(Xs + Xt-Xs)e-)
A /2((x,x) : x + x G • ) = ms(-) + p(-).
Suppose now that (3.13) and (3.14) hold, and let ms be a subsequential
limit such that n' F(a~?Xs G • ) -^ ms(-). By the same arguments as
above it follows that
ri F(a~}(X3 + Xt-Xs)e-)^ /2((x, x) : x + x G • ) = m3(-) + //(•)
However, n' F(a~}(Xs-\-Xt — Xs) G • ) A rnt(-) along every subsequence
(n') so we must have ms = mt —
p.
To prove Theorems 3.12 and 3.18 we need a couple of technical lemmas.
For e > 0, positive integer p and M C [0,1] we say that an element
x G D has e-oscillation p times in M if there exist to, , tp G M with
to < • • < tp such that |x^ — xij_11 > e for i = 1,... ,p. Let
B(p, e, M) = {x G D : x has e-oscillation p times in M}.
The following lemma is an immediate consequence of Lemma 2 p. 420
in Gihman and Skorohod [20].
88 Chapter 3. Regular variation for stochastic processes
Lemma 3.22 Let X = (Xt)iG[0,i] ^e a Markov process with sample
paths in D. If for e > 0 and 0 < Ti < T2 < 1 the quantity ctej^Ti2^T2 —
Ti) is less than 1, then
®e/4,T2(T2-Ti)F(XeB(l,e,[Ti,T2]))<
1 - Û!e/4,T2 (Î2 -Ti)
Proof. First,
P(XG5(l,e,[T1,T2])) = P( sup |X,-Xs|>e)s,te[TuT2]
< F( sup |Xs-XTl| >e/2),aG[Ti,T2]
and, by Lemma 2 p. 420 in Gihman and Skorohod [20],
P(|Xt -Xt I > e/4)P( sup |x. - XTJ > e/2) <-^_Zk Wt\-
Finally, P(|XTa - XTl| > e/4) < o;e/4)T2(T2 - 7\) from which the con¬
clusion follows. D
Lemma 3.23 LetX = (Xt)fG[0;i] be a strong Markov process with sam¬
ple paths in D such that ct;r)i(l) —y 0 as r —y oo. Suppose there exist
a sequence (an), 0 < an t °°; and Radon measures mo and mi on
#(ld\{0} with mo(Md\Md) = mi(Md\Md) = 0 such that
nF(a~1X0 e • ) 4m0(') and nF(a~1Xi e • ) ^y mi(-)
on B(Md\{0}). Then, for every e > 0, nP(X G anB(2,e, [0,1])) -y 0 as
n —y oo.
Proof. Fix an arbitrary e > 0 and let rn = inf{t : \Xt — X0| > ane/2}with the convention inf 0 = oo. Then
nP(Xean5(2,e,[0,l]))
< nE(l{rn<1}E^X- (lB(l,a„e,[r„,l])(X)))< nE(l{rn<1}aaTie/4)1(l)/(l - aane/4>1(l)))
0!ane/4,l(l)nP( sup |Xt -X0| > ane/2)
3.2. Asymptotically independent increments 89
by combining Lemma 3.22 and the strong Markov property. Moreover,
by combining Lemma 2 p. 420 in Gihman and Skorohod [20] and Lemma
3.10,
W iv y I >> /o\ <rnF(\Xi-Xo\>ane/4)
nP( sup |Xt-
X0|> ane/2) < t—
*G[0,1]-L -«ane/4,lUJ
" ml(BO,e/d-m0{BCQ,e/4),
as n —y oo, from which the conclusion follows. D
Proof of Theorem 3.12. Fix s,t eT with s < t. We will show that
there exists a unique vague limit msj such that nF(a~1(Xs, Xt) G • ) A
mSjt(-)- By repeating the procedure one can then show that, for any k G
N, there exists a unique vague limit mtl,...,tk, with mtlj...jtk(M \Mdk) =
0, such that nP(a"1(Xfl,..., Xifc) G j A mtu...,tk (•) ifh,...,tke T.
By Lemma 3.10,
nP(a"1(Xt - X8) G • ) A mt(-) - ms(-).
Clearly, there exist unique vague limits (Radon measures) mSjS and
m on £(R2d\{0}) with mS:S(M2d\M2d) = m(M2d\M2d) = 0 such that
nP(a"1(Xs,Xs) g • ) A m^(-) and nP(a"1(0,Xi - Xs) G • ) ^ m(-)
on B(M \{0}). By arguments similar to those in the proof of Lemma
3.10,
nP(a"1(Xs,Xi)G-) = nP(a"1((Xs, X5) + (0,Xt - X5)) G • )
A m8>8(-)+m(.)=:ma>t(-) on ,B(R2d\{0}).
Note that, by Lemma 3.23,
nF(w"(X,ô) > ane) <nF(Xe anB(2, e, [0,1])) - 0
as n —y oo. Hence, for any positive e and n there exists an no such that
nF(w"(X,ô) > ane) < n for any ô G (0,1) if n > no- Hence condition
(3.7) of Theorem 3.8 holds.
It remains to show that conditions (3.8) and (3.9) also hold. Fix arbi¬
trary e > 0 and n > 0. By Lemma 2 p. 420 in Gihman and Skorohod [20],
nF(w(X,[l-Ô,l))>ane) < nP( sup |X* - Xi_Ä| > ane/2)te[i-s,i]
<wP(|Xi-Xi_f| >ane/4)
l-Û!ane/4,l(<5)
90 Chapter 3. Regular variation for stochastic processes
By Lemma 3.10, for 1 - 5 G T
Jim^nPdXi - Xi_*| > ane/4) = mi(B^e/A) - mi-ô(BcQe/A).
Hence, by (3.16) there exists a <5 > 0, 1 — ô eT such that
r »/ fY f) niw wrwP(|Xi - Xi_j| > qwe/4)
lim sup nF(w(X, [1—
o, 1))>
ane)< hmsup
ri->-oo n-^oo 1 — aane/4,l (")
= ml(B0,e/4) ~ ml-s(Bo,e/4) < V,
and it follows that (3.9) holds. That (3.8) holds is shown by an almost
identical argument. The conclusion now follows by Theorem 3.8. D
Proof of Theorem 3.18. First note that B(2, e, [0,1]) is open
and, by Lemma 3.23, rimmf^«, nP(X G anB(2, e, [0,1])) = 0. By
assumption, liminfn_^00nP(X G anG) > m(G) for every open bounded
G e B(D). Hence m(B(2, e, [0,1])) = 0. Since e > 0 was arbitrary
it follows that m(B(2, e, [0,1])) = 0 for every e > 0 and hence also
m(Ue>0,ee®B(2, e, [0,1])) = 0. Since
Ue>0,eGQ5(2,e,[0,l])=(ö\JD)UVc,
it follows that m(Vc) < m(Ue>0,eGQ-B(2,e, [0,1])) = 0. Moreover, by
Theorem 3.3, there exist a > 0 and a probability measure a such that
(3.5) holds. Furthermore, by Theorem 3.7,
nP(a"1X1 G •) AmoTT^O on 5(Rd\{0}),
which holds if and only if there exists a probability measure o~i on
B(Sd~x) such that, for every x > 0, as u —y oo,
p(|Xi| > u)>* *,(•) onB(S )
holds, and &i is given by
,.mo7rf1({xGld\{0} : |x| > l,x/|x| G • })
al-\') = ZZÂ
moTrf^jxGE \{0} : |x| > 1})
3.3. Filtered Levy processes 91
We have
P(|X|00>gn,X/|X|00G-)
F(\X\00>an)
_
nP(g~1X e{xeD : |x|qq > 1,x/|x|qq G • })
nP(anXX e{xeD: |x|oo > 1})
_^m({x G £> : Ixloo > 1,x/|x|qq G • })
m({x G o : [XU > 1})
which necessarily is equal to a(-). Moreover,
<r({x Gfli:x = yl[Vil]iv G [0,1], y G S^1})
_
m({xe D : [xloo > 1} n V)
m({x G D : |x|oo > 1})
m({x G Ö : Ixloo > 1})1
ra({xG£>: Ixl«, > 1})
and
m(7T1~1({x G Ëd\{0} : |x| > l,x/|x| G • }))
m(7rr1({xGËd\{0}:|x|>l}))
m(7r1~1({xGRd\{0}: |x| > l,x/|x| G • }) fl V)
m(7rr1({x G ld\{0} : |x| > 1}) n V)
m({x6D:x = ylM,t)G [0,1], |y| > 1,y/|y| G • })
m({x6D:x = ylM,üe [0,l],|y|> 1})
_
m({xG £> : |x|qq > 1,x/|x|qq = yl[t,,i],^ G [0,1],y G • })~~
m({x G D : |x|oo > l}n V)= <t({xG£>i :x = ylKl],^G [0,l],yG-}).
The conclusion follows. D
3.3 Filtered Levy processes
In this section we will give another application of regular variation on
D by studying asymptotics of stochastic processes Y of the type
Yt= ( f(t,s)dX8, te [0,1], (3.19)Jo
92 Chapter 3. Regular variation for stochastic processes
where X is a regularly varying Levy process with sample paths of finite
variation. The idea here is that X is a regularly varying strong Markov
process satisfying arji(l) —y 0 as r —y 0, and that extremes for the
process Y are caused by one big jump in the process X. It turns out
that Y and Hf(X), where Hf : D —y D is defined below, have the same
regular variation limit measure and that Hf is sufficiently regular so
that Theorem 3.7 can be applied (we only need that Hf is positively
homogeneous on V - the set of step functions with one step). In this
way we can show that the process Y is regularly varying. Furthermore,
and equally important, we are able to explicitly compute the spectral
measure of such processes. In doing so we provide a natural way to
understanding the extremal behavior of such filtered regularly varying
additive processes. As a concrete example we will compute the spectral
measure of an Ornstein-Uhlenbeck type process driven by a regularly
varying Levy process (Example 3.25). Note that finite variation of the
sample paths of X allows us to define the integral in (3.19) in a pathwise
sense. As in the previous section, let
V = {xefl:x = yl[w,i], v G [0, l],y G Ed\{0}}.
Moreover, let Vo — V U {0} and let do denote the so-called Ji-metric
(see Billingsley [5] p. 112). For x G D define
M(x) 4 {z G Vo : d0(z,x) = inf{do(z,x) : z G V0}},
i.e. M(x) consists of the step functions in D with one step that are
closest to x. Note that for every xGDwe have M(x) ^ 0, see Lemma
3.26 below for details. Define \Ü : D —y Vq such that for x G D we take
\P(x) to be a unique element of M(x) chosen according to some arbitrary
criteria (e.g. of the elements of M(x) with earliest jump choose ^(x)as the one with biggest jump). For a nonzero and continuous function
/ : [0, l]2 - M define hf : V0 -^ D by
Mx)* = f f(t,s)dxs, te [0,1].Jo
Finally, define Hf = hf o\ü. Note that Hf is in general not continuous.
However, it is continuous on Vo, and this is sufficient when considering
integrators whose regular variation limit measure concentrates on V C
V0.
3.3. Filtered Levy processes 93
Theorem 3.24 Let X = (Xt)te[o,i] be a Levy process on Md. Suppose
that there exist a sequence (an), 0 < an f oo; and a nonzero boundedly
finite measure m on B(D) with m(D\D) = 0 such that, as n —y oo;
nF(a~lX G • ) A m(-) on B(D).
For a nonzero and continuous function f : [0, l]2 —y M, define the process
Y = (Yt)te[o,i] by Yt = f0 f(t, s)dXs. Then Y has sample paths in D
and, as n ^ oo,
nF(a~1Y e •) AmoHj1(- DD) onB(D).
To illustrate Theorem 3.24 we will now compute the spectral measure of
an Ornstein-Uhlenbeck type process driven by a regularly varying Levy
process.
Example 3.25 Let X = (Xt)te[o,i] be a Levy process on Md with sam¬
ple paths of finite variation. Necessary and sufficient conditions for
having sample paths of finite variation are that the generating triplet
(A, v, 7) satisfies A = 0 and either (i) j/(Rd\{0}) < oo or (ii) j/(Rd\{0}) =
oo and J|xi<lx^0 WK^x) < oo (see Sato [38] p. 140). Suppose there
exist a sequence (an), 0 < an t oo, and a nonzero Radon measure mi
with mi(ld\Rd) = 0 such that
nP(a-1X1 G • ) A mi(-) on ß(Rd\{0}).
Since X has stationary and independent increments, this implies that
nF(a;1Xt G • ) A tmi(-) on £(Rd\{0}) for every t G [0,1]. Let
Y = (Yt)te[o,i\ be an Ornstein-Uhlenbeck type process driven by X,
given by
Yt = [ e-d{t-s)dX8, 9>0, te [0,1Jo
Hence, by combining Theorems 3.3, 3.12 and 3.24, for every x > 0, as
u —y oo,
^\\ *oo> UX, Ï/ I
t» t ' j w —a / •> z?/n\>X a(-) «.BID,),
94 Chapter 3. Regular variation for stochastic processes
where a > 0 is the tail index of Xi and it follows that
a(.)=F({Ze-^-vh[VA](t),te[0,l]}e-),
where Z and V are independent, the distribution of Z is the spectral
measure of Xi and V is uniformly distributed on [0,1].
For the proof of Theorem 3.24 we will need the following results.
Lemma 3.26 M(x) ^ 0 for every x G D.
Proof. Fix x e D. For e G (0,1], define
Ke = {{y,v) :y G B0iSuPte[oA]lxtl,ve [0,1-e]}.
For some e G (0,1] we have
inf{d0(z,x) : z G V0} = inf{rf0(yl[v,i],x) : (y,v) G Ke).
Since Ke is compact and since (y, v) \-^y do(yl[w,i], x) is continuous there
exists (y*, v*) G Ke such that
mf{d0(y±[t,,i],x) : (y, v) e Ke} = d0(y*l[uV], x),
i.e. M(x) is nonempty. D
Lemma 3.27 Hf = hf o ^ is continuous on Vo-
Proof. We first show that \I/ is continuous on Vo and then that hfis continuous. Take xq G Vo and let (xn) be a sequence in D such
that do(xTC,xo) —y 0 as n —>• oo. By construction, <io(^(xn),xn) <
cfo(xo,xn). Since \I/(xo) = xo we have
d0(^(x„),^(x0)) = do(^(xn),x0)
< d0(^(xn),xn)+ c?o(xn,xo).
Hence do(^(xn), ^(xo)) —y 0 as n —y oo which proves the first claim.
We now show that hf is continuous. It is sufficient to show that hf
3.3. Filtered Levy processes 95
is continuous on Vo C D equipped with the Skorohod metric since this
metric and the Ji-metric are equivalent (see Billingsley [5] p. 114). Take
x G V and let (xn) be a Vo-valued sequence such that xn —y x. This
implies that there exists no such that xn G V for n > no and hence
we can without loss of generality assume that xn G V for every n.
Then there exist y,yn,v,vn such that xn = ynl[Un,i] and x = yl[v,i].Moreover, there exists a sequence (An) of strictly increasing continuous
mappings of [0,1] onto itself satisfying suptGr0 -n \Xn(t) — t\ —y 0 and
sup |ynl[ün,i](An(*))-yl[t,,i]WI->0 asn^oo.
te[o,i]
First we show that xn —y x implies that yn —y y and vn —y v. Since
sup \ynl[vn,i](K(t)) - yl[«,i] W| > |yn - y|,te [o,i]
it follows that yn —y y. Suppose that vn -ft v. Then there exists e > 0
such that limsup^^^ \vn — v\ > e. Since suptGr01i \Xn(t) — t\ —y 0 and
yn —>• y, this implies that
limSUp SUp |ynl[r,„,l](AnW)-yl[t;,l]WI ^ MAn-^oo te [0,1]
which is a contradiction. Hence vn —» v. We may now proceed to show
that xn —y x implies hf(x.n) —y hf(x). Indeed
sup
*e[o,i]
/»A„(i) pt/ f(\n(t),s)dXn(s)- / f(t,s)d*(s)Jo Jo
< SUp \f(\n(t),Vn)(ynl[Vn,l](\n(t))-yl[v,l](t))\te[o,i]
+ sup \(f(\n(t),vn)-f(t,v))yl[Vil](t)\*G[0,1]
< sup \f(Xn(t),vn)\ sup \ynl[Vn,i](\n(t)) - yl[v,i](t)\te[o,i] te[o,i]
+ |y| SUp \f(\n(t),Vn) ~ f(t,v)\.te [o,i]
Since / is bounded and xn —» x,
sup \f(Xn(t),vn)\ sup |ynl[t,n,i](An(i)) -yl[t,,i]W| ->• 0.
*G[0,1] *G[0,1]
96 Chapter 3. Regular variation for stochastic processes
Since / is uniformly continuous on [0, l]2, supfGr0jli \Xn(t) — t\ —>• 0 and
vn —y v it follows that
sup \f(Xn(t),vn)- f(t,v)\ ->0,*e[o,i]
i.e. hf is continuous on V. Finally, if xn —y 0, then clearly hf(x.n) —y 0.
Hence hf is continuous on Vo-
Lemma 3.28 If B G B(D) is bounded inD, then Hj1(Br\D) e B(D)is bounded in D.
Proof. We will show that for each r > 0 there exists an r = r(r, /) > 0
such that sup/G[0)1]|/0'/(*5s)d*(x)a| > r implies suptG[0)1] |xt| > r,
i.e. that HJ1(Bqt) C Bq ~,from which the conclusion follows. Fix
r > 0 and suppose that
sup | / /(i,s)d#(x)fl| >r.
*G[0,1] Jo
Then tf (x) = yl[„,i] with y G Rd\{0} and v G [0,1). Hence
te
which implies
sup | / f(t,s)d^(-x)s\ = sup \f(t,v)\\y\>rg[o,i] Jo te[o,i\
|y|>suPu,ve[o,i]\f(u,v)\'
Since, by construction of \I/, supfGr01] |x/| > |y|, we have
i ir
sup xt > —
-.
te[o,i] suPu,ve[o,i]\f{uiv)\
D
Lemma 3.29 Y has sample paths in D.
Proof. By assumption there exists fi'cfi with F(Q') = 1 such that
for each uj G Q', X(uj) G D and has finite variation. For such uj we also
3.3. Filtered Levy processes 97
have, since / is continuous on [0, l]2 and hence also uniformly continuous
on [0, l]2,
limv\t
/ (f(t,s)-f(v,s))dX8(u)'[0,v]
<lim sup \f(t,s)- f(v,s)\FV(X(u);[0,t]) = 0,vt* 8e[o,t]
where FV(g;T) denotes the total variation of g on T C [0,1]. Hence,
for uj e Q',
\im(Yt(uj) - Yv(uj)) = lim ( / /(*, s)dXs(uj) - [ f(v, s)dXs(u)VV vft \J[o,t] J[0,v]
= lim f (f(t, s) - f(v, s))dXs(uj) + lim f f(t, s)dXs(uj)vt* J[o,v] vtt J(vj]
= 0 + f(t,t)(Xt(uj)-Xt-(u)),
since X(uj) is right-continuous with left limits. Similarly,
\im(Yv(uj) - Yt(uj)) = lim ( / f(v, s)dXs(uj) - [ f(t, s)dXs(u)H* vit \J[o:V] j[o,f]
= lim f (f(v, s) - f(t, s))dXs(co) + lim f f(v, s)dX8(uj)H* J[0,t] vit J(t,v]
= 0 + f(t,t)(Xt+(uj)-Xt(u)) = 0.
Hence Y (a;) is right-continuous with left limits. D
Proof of Theorem 3.24. By Lemma 3.29, Y has sample paths in
D. Since m vanishes on Ve and, by Lemma 3.27, Hf is continuous on
V, it follows as in Theorem 3.7 that
nP(i7/(a~1X) G )AmoHj1(- DD) on B(D)
(here we do not need positive homogeneity of Hf). We now show that
this implies that nP(a~xY G • ) 4> m o Hjl( DD) on B(D), from
which the conclusion follows. Without loss of generality we assume
that supwvG[01] \f(u,v)\ — 1 (to avoid having to introduce additional
constants). For y G D and r > 0 let Byr = {z G D : d0(y,z) < r}.Fix arbitrary x G -D\{0} and 0 < e < ô < 7 with 7 + ô < cfo(x, 0)
98 Chapter 3. Regular variation for stochastic processes
and e < 7 — ô. Moreover, 7 and ô are chosen so that 5X,7, SXj7_j and
BXjl+s are mo H, -continuity sets. Let Xn be given by
X?^/*l«8....(AX.)dX.*J u
(where £0)a„e = {* G Md : |x| < ane}). Then Xn and Xn = X-XTC are
independent Levy processes for all n. Hence we can write Y — Yn+Yn
where Yn and Yn are independent and Y — J0 /(£, s)dX and Y —
/0'/(M)dX£. Note that
nP(a-1YG5X;7) > nF(a~1Yn e B^^s^'1^ e B0jS)
= nP(a-1Yn G B^_s)F(a-1Yn G So,*)
and that
nP(a-1YGSx,7) < nF(a~1Yn G 5x,7+(5, a~lYn G So,*)
+nP(a-1Y-Gß^)- nP(a-xYn G 5x,7+(5) Pfe1^ G B0,s)
-^nF(a-1Yn e Bc0jS)
Hence if we show that
nP(a-1Y"G^,)^0,
nF(a~1Yn G Sx>7_*) - m(Hj\B^_ô) n Sg>e),
nP(a-xY" G 5X>7+J) - m(^/-1(JBX;7+,) n B^e),
then we can let ô —y 0 (from which e —)> 0 follows) and conclude that
nP(a~1Y G -Bx,7) -^ mo Hjl(Bx^). Since x and 7 were arbitrary the
conclusion then follows since the m oif~^continuity sets Bxr C D\{0}
generate B(D) n D. We first show that nF(a~1Yn G 5^) -> 0. Note
that Zn defined by
Ztnâ sup |/(n,^)| / |dX?|u,ve[o,i] Jo
is a Levy process with jumps bounded by anesupuuGr0-n |/(w,v)| — ane
and that
sup |Yfn| < Z?. (3.20)*e[o,i]
3.3. Filtered Levy processes 99
Note that
vz?([anô,oo)) = u^(anB^s) = vy^1(an(B^ô n 50,e)) = 0
since e < ô (^Xn, ^Xn and i/Xi denotes the Levy measures of Zf, X
and Xi respectively, and Bq5,Boj G Hd). Hence, by Theorem 2.3,
nP(a-%n > Ô) -> 0 and hence, by (3.20), nP(fl-1Yn G Bfo) -> 0.
We now show that nF(a'lYn G B^7_s) -> m(HJ1(B^1_5) n 5g>e).First note that
nP(a-1YnG5X;7_,)- nF(a~1Yn G 5x>7-*,Xn g 5(2, ane, [0,1]))
+ nP(a-xYn G 5x>7_Ä,Xn G ß(2,aTCe, [0,1]))
- nP(a-1if/(X") G Bx>7_*,Xn g 5(2, ane, [0,1]))
+ nP(a-xYn G Sx>7_*,Xn G B(2,ane, [0,1]))
- nP(F/(a-1Xn) G Bx>7_*,Xn g B(2,ane, [0,1]))
+ nP(a-xYn G 5X,7_,,XTC G 5(2,aTCe, [0,1]))
(using the fact that Hf is positively homogeneous on V). Note that
nF(Hf(a~1Xn) e B^_s)
= nF(Hf(a~lXn) G Sx,7_j,Xn £ B(2,ane, [0,1]))
+ nP(ff/(a-1Xn) G ßx,7-*,Xn G 5(2, ane, [0,1])).
Since nP(a~1Xn G • ) -^ ra( • H Bq e) applying Theorem 3.7 yields
nF(Hf(a~1Xn) G Sx>7_j) -> m^1^^) n SJ>e).
Moreover,
nPtff/tû-1^) G Sx>7_j,Xn G 5(2,ane,[0,l]))
<nP(Xeß(2,an6,[0,l]))^0,
where the latter convergence follows from Lemma 3.23. Hence
nF(Hf(a-1Xn) G Bx>7_*,Xn g 5(2,ane, [0,1]))
->m(#71(£x>7-*)nBS,e)-
100 Chapter 3. Regular variation for stochastic processes
Moreover, by Lemma 3.23,
nF(a-1Yn G Sx>7_*,Xn G 5(2,ane, [0,1]))
< nP(X G 5(2,ane, [0,1])) -> 0.
Hence nP(a;1Yn G 5X,7_5) -> m(Hj1(B^1_ô) n 5£ e). By the same
arguments it follows that nF(a~ Yn G 5Xj7+<s) —y m(HJ (5X;7+j) D
3.3. Filtered Levy processes 101
o
I
00 02 04 06 08 1 0
Figure 3.1: 8 simulations of X \ {\XS\ > FT^. ,(0.9) some s G [0,1]},where X is a Levy process with X\ ~ C(l) (a Cauchy distribution with
density fXl(x) = 1/(tt(1 + x2))).
Chapter 4
Dependence in elliptical
distributions
This chapter and the following chapter form the second part of the the¬
sis. In these two chapters we study properties of elliptical distributions.
The class of elliptical distributions provides a rich source of multivari¬
ate distributions which share many of the tractable properties of the
multivariate normal distribution and enables modelling of multivari¬
ate extremes and other forms of nonnormal dependences. The general
representation theorem (Theorem 4.2) allows us to explicitly compute
various interesting quantities and dependence measures without having
to fix a particular elliptical distribution. Moreover, the representation
theorem provides us with a powerful tool for illustrating many different
dependence concepts for nontrivial multivariate models. This chapter is
organized as follows. In Section 4.1 we introduce the class of elliptical
distributions and recall some of its most important properties. We also
prove (Theorem 4.10) that sums of elliptically distributed random vec¬
tors with the same dispersion matrix are elliptical if they are dependent
only through their radial parts. This result has interesting applications
to multivariate time series. In Section 4.2 we study the concordance
measures Kendall's tau (r) and Spearman's rho (qs)- It is easily shown
103
104 Chapter 4. Dependence in elliptical distributions
that for a bivariate normal distributed random vector with linear cor¬
relation coefficient g the relation
2r — — arcsm g,
IT
between Kendall's tau and the linear correlation coefficient, holds. We
show that this relation holds more generally (subject to only slight mod¬
ifications), see Theorem 4.14 below, for all nondegenerate elliptical dis¬
tributions. One prime application of this result is robust estimation
of linear correlation coefficients for nonnormal elliptical distributions.
Monte Carlo studies indicate that this estimator performs better than
most of its competitors (see Figure 4.1 for an illustration). One might
also expect Spearman's rho to be invariant in the class of elliptical dis¬
tributions with continuous marginals and a fixed dispersion matrix. We
give a counterexample showing that this is not true.
4.1 Elliptical distributions
In this section we introduce the class of elliptically distributed random
vectors and give some of their properties. For further details about
elliptical distributions we refer to Fang, Kotz and Ng [17] and Cambanis,
Huang and Simons [11].
Definition 4.1 If X is a d-dimensional random (column) vector and,
for some vector fi G Md, some d x d nonnegative definite symmetric
matrix S and some function (f) : 1R+ —y M, the characteristic function
V?x-M of X —
/i is of the form <^x-M(t) = 0(tTSt); we say that X
has an elliptical distribution with parameters \i, S and (j), and we write
X~Ed(fjL,Z,(f>).
The function 0 is referred to as the characteristic generator of X. When
d = 1, the class of elliptical distributions coincides with the class of one-
dimensional symmetric distributions.
For elliptically distributed random vectors, we have the following general
representation theorem.
4.1. Elliptical distributions 105
Theorem 4.2 X ~ Ed(/J,,T>,(f)) with rank(S) = k if and only if there
exist a random variable R > 0 independent of XJ, a k-dimensional
random vector uniformly distributed on the unit hypersphere §2_ =
{z G Mk | zTz = 1}; and a d x k matrix A with AAT = S; such that
X = /i + RAV. (4.1)
For the proof of Theorem 4.2 and details about the relation between R
and 4>, see Fang, Kotz and Ng [17] or Cambanis, Huang and Simons [11].
Remark 4.3 (a) Note that the representation (4.1) is not unique: if Ö
is an orthogonal k x k matrix, then (4.1) also holds with A' = AÖ and
U' = oTu.
(b) Note that elliptical distributions with different parameters can be
equal: if X ~ F<f(/i, £,0), then X ~ Ed(fi, cS,0c) for every c > 0,
where 4>c(s) = 4>(s/c) for all s > 0.
Example 4.4 Classical examples of elliptical distributions are the mul¬
tivariate normal and the multivariate t-distributions. Let X = \i +
RAXJ ~ F^(/i,S,0), where rank(S) = d. Then X is normally dis¬
tributed if and only if R2 ~ Xd {Xd denotes a Chi Square distribution
with d degrees of freedom), and X is t-distributed with v degrees of free¬
dom if and only if R2/d ~ F(d, u) (F(d, u) denotes an F-distribution
with d and u degrees of freedom).
If the elliptically distributed random vector X has finite second mo¬
ments, then we can always find a representation such that Cov(X) = S.
To see this we use Theorem 4.2 to obtain
Cov(X) = Cov(/z + RAU) = AE(R2) Cov(U) AT,
i.e. Cov(X) exists if and only if K(R2) < oo. To compute Cov(U),let Y ~ Md(0,ld) and let | • |2 denote the Euclidean 2-norm. Then
Y = |Y|2U, where |Y|2 and U are independent. Furthermore |Y|| ^^
Xd, so IE(|"V^12) = d. Since Cov(Y) = Id we see that if U is uni¬
formly distributed on the unit hypersphere in Md, then Cov(U) = Id/d.
106 Chapter 4. Dependence in elliptical distributions
Thus Cov(X) = AAT~E(R2)/d. By choosing the characteristic generator
(f)*(s) = (f)(s/c), where c = K(R2)/d, we get Cov(X) = E.
The following result provides the basis of most applications of elliptical
distributions.
Lemma 4.5 LetX ~ Ed(fi, S, (ft), let B be aqxd matrix and letb G Mq.
Then
b + BX~ Eq(b + Bn, BZBT, 0).
Proof. By Theorem 4.2, b + BX has a stochastic representation
b + BX ± b + Bfi + ÜBAU
and the conclusion follows from Definition 4.1. D
If we partition X, \i and S into
where Xi and Hi are r x 1 vectors and Sn is a r x r matrix, then we
have the following consequence of Lemma 4.5.
Corollary 4.6 Let X ~ Ed(ß, S, (p). Then
Xi ~ Er(/ii, Sn, 0), X2 ~ Ed_r(/i2, S22, 0).
Hence, marginal distributions of elliptical distributions are elliptical and
of the same type (with the same characteristic generator).
Next we introduce the linear correlation coefficient for a pair of random
variables with a joint elliptical distribution.
Definition 4.7 Let X ~ Fd(/i,S,0). Fori,j G {l,...,d}, if Y^ > 0
and Tijj > 0, then we call
the linear correlation coefficient for (Xi,Xj)T.
4.1. Elliptical distributions 107
Note that if Var(JQ), Var(X,) G (0,oo), then
Qij = Cov(Xi,Xj)/^Vai(Xl)Vai(Xj),i.e. the linear correlation coefficient as defined by (4.2) is an exten¬
sion of the usual definition in terms of variances and covariances. We
want to interpret the linear correlation coefficient as a scalar measure
of dependence and, as such, it should not rely on finiteness of certain
moments. Clearly (4.2) only makes sense for elliptical distributions. On
the other hand, linear correlation is not always a meaningful measure
of dependence for nonelliptical distributions, whereas Kendall's tau and
Spearman's rho (discussed below) remain meaningful; see for example
Embrechts, McNeil and Straumann [16] p. 25.
A random variable is said to be continuous if its distribution function is
continuous. We now present necessary and sufficient conditions for the
components of an elliptically distributed random vector to be continuous
random variables.
Lemma 4.8 Let X ~ Ed(/i, £,</>), with P(JQ = fii) < 1 for all i G
{1,..., d} and with representation X = fi + RAXJ according to Theorem
4-2. //rank(S) = 1, then Xi,... ,Xd are continuous random variables
if and only if R is continuous. If rank(S) > 2, then Xi,... ,Xd are
continuous random variables if and only ifF(Xi = fii) = 0 for all i, or
equivalently, if and only ifF(R = 0) = 0.
Proof of Lemma 4.8. Let X = fi + RAXJ be a stochastic repre¬
sentation according to Theorem 4.2. Suppose rank(S) = 1, then A
is a d x 1 matrix and U is symmetric {1,—1}-valued. Furthermore,
F(Xi = fii) < 1 implies An ^ 0. Hence, if rank(S) = 1, then Xi,..., Xd
are continuous random variables if and only if R is continuous. Now
suppose rank(S) = k > 2. Define A; = (An,..., A^) and a = AiAj.Since F(Xi = fii) < 1, the case a = 0 is excluded. By choosing an
orthogonal k x k matrix O whose first column is Aj/a and using Re¬
mark 4.3(a) if necessary, we may assume that A^ = (a, 0,..., 0), hence
Xi = fii + aRUi. Note that Ui is a continuous random variable be¬
cause k > 2. Hence F(aRUi = x) = 0 for all x G M\{0}. Hence, if
108 Chapter 4. Dependence in elliptical distributions
rank(S) > 2, then Xi,..., Xd are continuous random variables if and
only if F(Xi = fii) = 0 for i = 1,..., d, or equivalently, if and only if
F(R = 0) = 0. D
The following lemma states that linear combinations of independent
elliptically distributed random vectors with the same dispersion matrix
S (up to a positive constant, see Remark 4.3) remain elliptical.
Lemma 4.9 Let X ~ Ed(fi,^,4>) and X ~ Ed(ß, cS,0) for c > 0 be
independent. Then for a,b G M, aX + &X ~ Ed(a/i + bfi, S, (j)*) with
4>*(u) = (j)(a2u)^(b2cu).
Proof. For all t G Md,
= 0((at)TS(at)) 0((&t)T(cS)(6t))= 0(a2tTSt)0(&2ctTSt).
The next theorem shows that this remains true if we allow the elliptically
distributed random vectors to be dependent only through their radial
parts.
Theorem 4.10 Let R and R be nonnegative random variables and let
X = /z + flZ ~ Ed(fi,T1,4)) and X =jl + KL ~ Fd(/I,S,0)? where
(R, R),Z,Z are independent. Then X+X ~ Ed(fi+Ji, S, 0*). Moreover,
if R and R are independent, then 4>*(u) = <fi(u)(f)(u).
For the expression of the characteristic generator, <fi*, we refer to the
proof below.
Proof. Let (f)^r> be the characteristic generator of (R \ R = r)Z, let
4>' be the characteristic generator of Z, and let Fr be the distribution
function of R. Then for all t G
•>oo
cf>'(r2tTXt)ftr\tTXt)dFR(r),
'0
4.2. Kendall's tau and Spearman's rho 109
from which it follows that X + X ~ Ed(fi + fi, S, (j)*), with
<f>*(u) = / (p'(r2u)^r)(u)dFR(r).Jo
Moreover, if R and R are independent, then (p^r\u) = (j)(u) and
(f>*(u) = / (p'(r2u)(j){r)(u)dFR(r)JO
= (f>(u) (p'(r2u)dFR(r) = (f)(u)(p(u).Jo
D
A natural application of Theorem 4.10 is in the context of a multivariate
time series.
Example 4.11 Let Xt = crtTt, t G Z, where the random vectors
Tit ~ Ed(0, S, (pt) are mutually independent and independent of the non-
negative (univariate) random variables at for all t. The cr^'s are allowed
to be dependent. Then for every t G Z, Xt is elliptically distributed
with dispersion matrix S, and so are all partial sums Sn = ^"=1 ^-t- '
4.2 Kendall's tau and Spearman's rho for
elliptical distributions
To begin with we recall the definitions of the concordance measures
Kendall's tau and Spearman's rho. For more on the properties of con¬
cordance measures and in particular on Kendall's tau and Spearman's
rho we refer to Joe [25] and Nelsen [30] and the references therein.
Definition 4.12 Kendall's tau for the random vector (Xi,X2)T is de¬
fined as
t(Xi,X2)±F((Xi-X[)(X2-X'2)>0)-F((X1-X[)(X2-X'2)<0),
where (X[,X2)T is an independent copy of (Xi,X2)T.
110 Chapter 4. Dependence in elliptical distributions
Definition 4.13 Spearman's rho for the random vector (Xi,X2)T is
defined as
Qs(Xi,X2) â 3(p((Xi - X[)(X2 - X'i) > 0)
-F((Xi-X[)(X2-X'2')<0)),where (X[,X2)T and (X'{,X2)T are independent copies of (Xi,X2)T.
An important property of Kendall's tau and Spearman's rho is that
they are invariant under strictly increasing transformations of the un¬
derlying random variables. If (Xi,X2)T is a random vector with con¬
tinuous univariate marginal distributions and Ti and T2 are strictly in¬
creasing transformations on the range of X\ and X2 respectively, then
t(Tx(Xi),T2(X2)) = t(Xi,X2). The same property holds for Spear¬
man's rho. Note that this implies that Kendall's tau and Spearman's
rho do not depend on the (marginal) distributions of X\ and X2.
The following theorem relates Kendall's tau and the linear correlation
coefficient for two random variables with a joint elliptical distribution.
Its proof is a combination of Lemmas 4.17 and 4.22 below. For the case
of normal distributions, see also Lemma 4.21.
Theorem 4.14 Let X ~ Ed(fi,^,(p). If for iJ G {l,...,d}, F(Xi =
fii) < 1 and F(Xj = fij) < 1? then
T(Xt,Xj) =(l- ^2(¥(Xi = x))2) 1arcsin^-, (4.3)
^xem. '
where the sum extends over all atoms of the distribution of Xi. If in
addition rank(S) > 2, then (4.3) simplifies to
r(Xi,Xj) = (1 - (F(Xi = fn))2) - arcsin^-, (4.4)7T
which further simplifies to
2r(Xi, Xj) = — arcsin Qi~ (4-5)
7T
ifF(Xi = fil) = 0.
4.2. Kendall's tau and Spearman's rho 111
Example 4.15 Let X ~ Ed(fi,T,,(j)), where rank(E) > 2 and F(X{ =
fii) = 0 for every i. Let H denote the distribution function of X and
let Fi denote the distribution function of Xi. If Fi,..., Fd are arbitrary
continuous distribution functions and for every i, F~l(u) = inî{x G M :
Fi(x) > u} with the convention inf 0 = oo, then
XA(F-1(Fi(X1)),...,F-\Fd(Xd)))T
has univariate marginals Fi,...,Fd and r(Xi, Xj) = r(Xi,Xj). In par¬
ticular, gij = sin(7TT(Xi, Xj)/2) which means that Theorem 4.14 en¬
ables parameterization of multivariate models constructed by marginal
transformations of elliptical distributions.
As a consequence of Theorem 4.14 we have the following well-known
result for Spearman's rho, for which we give an easy proof for complete¬
ness.
Corollary 4.16 LetX ~ Afd(fi, E); where fori, j G {1,..., d}, E^ > 0;
Hjj > 0. Then
gs(Xi,Xj) = - arcsin(gij/2). (4.6)7T
Proof. Recall that g^ = E^-/^/E^E^-. Let Xi = Xi for i = 1,..., d
be mutually independent, and independent of X. Then X ~ Nd(fi, E),where E = diag(En,..., Edd). Hence, X* = X-X ~ Afd(0, E*), where
E* = S + E. Let q\. â E*./V/Ë5Ë*-. Then,
gs(Xi,Xj) = 3t(X*,X*) =31— arcsinp^ J = - arcsin(^/2),
where the second equality follows from Theorem 4.14 and the fact that
the dispersion matrix of a sum of two independent identically distributed
elliptical random vectors differs from those of the terms by at most a
positive constant factor. D
In the light of Theorem 4.14 one might expect Spearman's rho to be in¬
variant in the class of elliptical distributions with continuous univariate
112 Chapter 4. Dependence in elliptical distributions
marginals and a fixed dispersion matrix. However, the counterexample
below shows this to be not true.
Counterexample. Let X ~ M2(fi, E), where En, E22 > 0. According
to Theorem 4.2, X has a stochastic representation X = fi-\-RAXJ, where
R ~ xi- We construct a counterexample by deriving a relation between
Spearman's rho and the linear correlation coefficient for the bivariate
elliptically distributed random vector W = AXJ. The relation is given
by
,rrr rrr.
„
/arcsino\t
/arcsino\es(wi.wâ) = 3(—^J-4(—r-^J ,
where g = Ei2/v/^ii^22- For a proof, see Section 4.4. This relation
differs from the relation (4.6) between Spearman's rho and the linear
correlation coefficient for a bivariate normal distribution. The differ¬
ence gs(Xi,X2) — gs(Wi,W2) as a function of the linear correlation
coefficient g is plotted in Figure 4.2. It should be noted that there
are other choices of R (other than R? ~ xi) ^OI which the difference
gs(Xi,X2) — gs(Wi, W2) becomes much bigger.
4.3 Proof of Theorem 4.14
The following lemma gives the relation between Kendall's tau and the
linear correlation coefficient for elliptical random vectors of pairwise
comonotonic or countermonotonic components. It proves Theorem 4.14
for the case rank(E) = 1.
Lemma 4.17 Let X ~ Ed(fi, E, (p) with rank(E) = 1. If F(Xi = fii) <
1, and F(Xj = fij) < 1, then
r(X,,Xj) =(l- ^2(¥(Xi = x))2) 1 arcsin^-. (4.7)^
xem.'
Proof. Let X be an independent copy of X. Let X = fi + RAU and
X = fi + RAU be stochastic representations according to Theorem 4.2,
4.3. Proof of Theorem 4.14 113
where (R, Ü) denotes an independent copy of (R, U). In particular, A
is a d x 1 matrix and U is symmetric {1,—1}-valued. Furthermore,
F(Xi = fn) < 1 and F(Xj = fij) < 1 imply An / 0 and AjX / 0.
Therefore,
Qij = AiiAjl/^JA\A2X = sign^i^ji) = - arcsin^-, (4.8)
(Xi - Xi)(Xj - Xj) = AitAji(RU - RÜ)2 and
F(RU = RÜ) = ^2(F(RU = x))2 = ^2(F(Xi = x))2. (4.9)
If AnAji > 0, then by Definition 4.12
r(Xi,Xj) = F((RU - RÜ)2 > 0) = 1 - F(RU = RÜ)
Using (4.8) and (4.9), the result (4.7) follows. If AiïAjl < 0, then
r(Xi,Xj) = -F((RU - RÜ)2 > 0)
and the result (4.7) follows in the same way. D
Lemma 4.18 Let X ~ Ed(fi, E,0) with rank(E) = k > 2 and let X
be an independent copy of X. If F(Xi = fii) < 1, then F(Xi = Xi) =
(F(Xt = fn))2.
Proof. Let X = fi + RAXJ be a stochastic representation according
to Theorem 4.2. Define A; = (An,... ,A^) and a = AiAj. Since
F(Xi = fii) < 1, the case a = 0 is excluded. By choosing an orthogonal
k x k matrix Ö whose first column is Aj/a and using Remark 4.3(a) if
necessary, we may assume that A^ = (a, 0,..., 0), hence Xi = fii+aRUi.Note that Ui is a continuous random variable because k > 2. Hence
F(aRUi =x)=0 for all x G M\{0}, and it follows that
F(Xi = Xi) = ^2(¥(Xi = x))2 = £)(P(aÄ*7i = x))2 = (F(Xi = fit))2xei set
Lemma 4.19 Let X ~ Ed(fi, E,0) with rank(E) = k > 2, and let X
be an independent copy of X. If F(Xi = fii) < 1 and F(Xj = fij) < 1,
then
r(Xt,Xj) = 2P((X, - Xi)(Xj - Xj) > 0) - 1 + (F(Xt = fn))2. (4.10)
114 Chapter 4. Dependence in elliptical distributions
Proof. Since Y = X — X ~ Fd(0,E,^>2), there exists a stochastic
representation Y = RAXJ according to Theorem 4.2. By Lemma 4.18,
F(Yi = 0) = (F(Xi = fn))2 < 1 and similarly F(Yj = 0) < 1. Define
Ai = (An,... ,Aik) and Aj = (Aji,... ,Ajk). With the same argu¬
ments as in the proof of Lemma 4.18, it follows that A;U and AjXJ are
continuous random variables, which implies that P(A;U = 0) = 0 and
P(AjU = 0) = 0. Therefore,
F(YiYj = 0) = P(Ä = 0) = F(Y, = 0) = (¥(Xi = fi^)2.
Since r(Xi,Xj) = 2¥(YiYj > 0) - 1 + F(YiYj = 0), the conclusion
follows. D
Lemma 4.20 Let X ~ Ed(0, E,0) and X ~ Fd(O,cE,0) with c > 0
and rank(E) >2. If ¥(Xt = 0) < 1 and F(X{ = 0) < 1, then
F(XiXj > 0)(1 - F(Xt = 0)) = ¥(XiXj > 0)(1 - P(Xj = 0)).
Proof. Take X = RAXJ according to Theorem 4.2 and set W = AU.
Then
F(XiXj > 0) = F(RWiRWj > 0)
= F(RWiRWj > 0 | R > 0) F(R > 0)
= P(WiWi > 0)F(R>0).
Furthermore, X = a/cäW according to Theorem 4.14, and a similar
calculation shows
F(XiXj > 0) = F(cR2WiWj > 01R > 0) ¥(R > 0)
= P(^W7- >0)P(A>0).
As in the proof of Lemma 4.18, it follows that Wi has a continuous
distribution. Therefore, ¥(R > 0) = 1 - ¥(Xt = 0) and ¥(R > 0) =
1 - F(Xi = 0), and Lemma 4.20 follows. D
Although the next result for normal distributions is well known, we give
a proof for completeness of the exposition and for showing where the
arcsin comes from.
4.3. Proof of Theorem 4.14 115
Lemma 4.21 Let X ~ JVd(/z,E). // ¥(X{ = fii) < 1 and F(Xj
fij) < 1, then
2
r(Xi,Xj) = 2F((Xi - Xi)(Xj - Xj) > 0) - 1 = - arcsin^-,7T
where X is an independent copy of X.
Proof. Using ai = \/E^ > 0, Oj = sj T>jj > 0 and Qij = Ejj/cr^crj, we
have
ynj _ [^H ^U ] _ [ Gi Vi^jQij \
Define Y = X - X and note that (Yi, Yj) ~ Af2(0, 2Hij). Furthermore,
(Yi,Yj) = \l/2(ö-iVcos^j + aiW sin (fij, ajW), where ifij = arcsin^j G
[—7r/2, 7t/2] and (V, W) is standard normally distributed. By the radial
symmetry of (Yi, Yj),
r(Xi,Xj) = 2P(riyi>o)-i= 4P(F; >0, Yj >0)-l
= 4F(V cos <fij+ W sin (fij >0,W >0)-l.
If $ is uniformly distributed on [—ir, tt), independent ofR= \/V2-\- W2,then (V, W) = Ä(cos $, sin$) and
r(Xi,Xj) = 4 P(cos $ cos ifij + sin $ sin (fij > 0, sin $ > 0) - 1
= 4 P($ G ((fij - tt/2, ifij + tt/2) n (0, tt)) - 1
ifjj + tt/22tt
which simplifies to (2/7r) arcsin ^-. D
Lemma 4.22 Let X ~ Ed(fi,Z,(p) with rank(E) = k > 2. If F(X{ =
fii) < 1 and F(Xj = fij) < 1, then
r(Xi,Xj) = (1 - (F(Xi = fn))2) 1 arcsin^-. (4.11)
116 Chapter 4. Dependence in elliptical distributions
Proof. Let X be an independent copy of X. By Lemma 4.19, we can
use (4.10). By Lemmas 4.9 and 4.18, X-X ~ £d(0, E, 0*) withPpQ =
Xi) = (F(Xi = fii))2 < 1 and F(Xj = Xj) = (F(Xj = fij))2 < 1. If Z,
Z ~ Nd(fi, ^]/2) are independent, then Z — Z ~ A/d(0, E). By Lemma
4.20,
F((Xi-Xi)(Xj-Xj) > 0) = ¥{{Zi-Zi){Zj-Zj) > 0)(l-(F(Xi = fn))2).
Substituting this into (4.10) and using Lemma 4.21, the result (4.11)follows. D
4.4 Proof of the counterexample
In this section we give a more detailed version of the counterexample
already discussed in Section 4.2.
Counterexample. Let X = fi+RAXJ ~ E2(fi, E, cp), where En, E22 >
0 and fi, R, A and U are as in Theorem 4.2. To construct a counterexam¬
ple we derive the relation between Spearman's rho and the linear corre¬
lation coefficient g = Ei2/VEiiE22 for W = AU. We only consider the
case with rank(E) = 2, since the case with rank(E) = 1 is trivial. From
the invariance of Spearman's rho under componentwise strictly increas¬
ing transformations of the underlying random vector we can without
loss of generality assume that En = E22 = 1 and Ei2 = E2i = g. We
show that the following relation holds,
,rrr rrr.
„
/arcsino\t/arcshifA
., _,
Qs(Wi,W2) =
3{—^-j -4(—^-j (4.12)
In the case of a bivariate normal distribution, i.e. R ~ x%-> we know
from Corollary 4.16 that the relation between Spearman's rho and the
linear correlation coefficient is
Qs(XuX2) = -MC8m(Q/2). (4.13)7T
Since these two relations differ (the difference is plotted in Figure 4.2)we conclude that, contrary to Kendall's tau, Spearman's rho is not
4.4. Proof of the counterexample 117
invariant in the class of elliptical distributions with a fixed dispersion
matrix. It remains to be shown that (4.12) holds. This can be done
following the steps below.
Step 1. Let (Wi,W2), (W[, W£) and (W[', Wl[) be independent copies.
Then
gs(Wi,W2) = 12F(W[ < Wi,W2 < W2) - 3.
Step 2. For (Wx, W2), W[, W2' as above we have that,
F(W[ < Wi,W2 < W2)
1 f27r fl 1. , . ,
= — / —I—arcsm(sin arcsmg + t )27tJq \2 TT
V V "
1arccos cost
2ttv ;
r arccos(cost) arcsin(sin(arcsin g + t)) ) dt.7TZ J
Step 3. The following equalities hold:
/>2tt
(i) / arcsin(sin(arcsin g + t))dt = 0.Jo
/»2tt
(ii) / arccos(cosi)dt = 7T2.Jo
(iii) / arccos(cos t) arcsin(sin(arcsin g + t))dtJo
2t• ^3
^•
= -( arcsm g) arcsm g.
3VH)
2H
Combining Steps 1-3 yields (4.12),
gs(Wi,W2) = 12F(W[ < Wi,W2' < W2) - 3
12 / tt 2, . xo
1.
\0
= — 7T tt (arcsmg) -\— arcsm g — 3
2tt V 2 3tt2v *'
2 V
/arcsinö\ /arcsino\=
\-^r)-\-ir)
Proof of Step 1. Straightforward computations of Spearman's rho
118 Chapter 4. Dependence in elliptical distributions
for continuous random variables yields
Qs(Wi,W2) = 3(2F((Wi-W[)(W2-W%)>0)-1)
= 3 (4F(W[ < Wi,W2 < W2) - 1)
= 12F(W[ < Wi,W2 < W2)-3.
D
Proof of Step 2. Let (p,(p',(p" ~ U(0,27r) be independent. Then
(Wi,^) = (cos <£?, sin(arcsin £> + ip)),
(W[,W2) = (cos^/,sin(arcsin^ + (^/)),
(W", W2) = (cos if", sin(arcsin g + f")).
Since
F(W[ < Wi,W2' < W2)
= P(cos if' < cos if, sin(arcsin g + if") < sin(arcsin g + if)),
conditioning on if yields,
F(W[ < Wi,W2 < W2)
i r2ir— / P(C0S t - COS if' > 0)2ttJo
2tt
2î~wu
P(sin(arcsin g + t) — sin(arcsin g + if") > 0)dt.
The factors in the integrand can be written as
P(cosi — cos if' > 0) = 1 2arccos(cost)27T
and
P(sin(arcsin g + t) — sin(arcsin g + if") > 0)
= 1 — — (tt — arcsin(sin(arcsin g + t)) — arcsin(sin(arcsin g + t)))
= —| 2 arcsin(sin(arcsin g + t)).2 27T
4.4. Proof of the counterexample 119
Combining these expressions yields,
P(cost — cos f' > 0) P(sin(arcsin g + t) — sin(arcsin g + if") > 0)
= - H— arcsin(sin(arcsin g + t)) — —— arccos(cost)2 7T 27T
arccos(cost) arcsin(sin(arcsin g + t)).
D
Proof of Step 3. (i) and (ii) are elementary. To compute (iii) we
first split the integral depending on arccos(cost) and then use a variable
transformation to obtain
1 f2n
1 =— / arccos(cos t) arcsin(sin(arcsin g + t))dtTT2,KJO
i ( r=—- / t aicsm(sin(avcsin g + t))dt
71-2 \Jo
f2n \+ / (27T — t) arcsin(sin(arcsin £> + t))dt J
1 / A'^'+a'rcsin g
(u — arcsin g) arcsin(sinu)duTT2
,
,"l
-' arcsin g
/»27r+arcsin g
+ / (27T — u + arcsin g) arcsin(sinu)duJ 7r+arcsin g
120 Chapter 4. Dependence in elliptical distributions
Hence,
/
7TZ/ (u — arcsin g) arcsin(sin u)du/o
V—; ;'
/»arcsin g
- (u — arcsin g)uduJo
II
/»7r+arcsin g
+ (u — arcsin g)(it — u)dun
s.
III
/»2tt
+ / (27T — u + arcsin g) arcsin(smw)dwJ n
IV
/»7r+arcsin g
/ (27T — u + arcsin g) (w — u)dun
V
/»27r+arcsin^
+ / (27T — u +arcsin g)(u — 27t) duJl-Ks
v 7VI '
The different parts can now be computed separately.
fTÏ /»7T/2
1=1 (u — arcsin g) arcsin(sinu)du = / (u — arcsin g)uduJo Jo
+ I (u — arcsin g)(it — u)du = 7r3/8 — 7r2 (arcsin g)/AJ-k/2
/»arcsin g
II = / (u — arcsin g)udu = — (arcsin g)3/QJo/»7r+arcsin g
III = / (u — arcsin g)(n — u)duJ TV
= — 7r(arcsin g)2/2 + (arcsin g)3/6
4.4. Proof of the counterexample 121
/»Z7T
IV = / (27T — u + arcsin g) arcsin(sinu)duJ n
/»3?r/2= / (27T — u + arcsin g) (it — u)du
J n
/>2tt
+ / (27T — u + arcsin g) (u — 2-7r)dwJz-k/2
= - 7r3/8 - 7T2(arcsin g)/A/»7r+arcsin g
V = / (27T — u + arcsin g) (it — u)duTV
= — 7T (arcsin g)2/2 — (arcsin g)3/G/»27r+arcsin g
VI = / (27T — u + arcsin g) (u — 2it)du = (arcsin g)3/Q
Putting everything together yields
1 = \(I-II + III + IV -V + VI)7TZ
\ (1,. sQ
TT2
TT"2" V3=
—g ( - (arcsin £>)3 —— arcsin £>
2, . ^ 1(arcsin g) — - arcsin g.
3tt2Vu/
2
D
122 Chapter 4. Dependence in elliptical distributions
Standard Estimator
0 500 1000 1500 2000 2500 3000
Kendall's tau Transform
0 500 1000 1500 2000 2500 3000
Figure 4.1: Linear correlation estimates for 3000 independent samples
of size 90 from a bivariate t$-distribution with linear correlation coeffi¬
cient 0.5. The lower figure shows linear correlation estimates using the
estimator sin(7rr/2) where r denotes the Kendall's tau estimator.
4.4. Proof of the counterexample 123
0.003
0.002
0.001
Figure 4.2: The difference between (6/7r) arcsin(^>/2) and
(3/V) arcsin g— (4/7T3) (arcsin £>)3 as a function of g for g G [0,1]
(see the counterexample in Section 4-2).
Chapter 5
Multivariate extremes
for elliptical distributions
In this chapter we analyze multivariate extremes for elliptical distri¬
butions. The analysis also highlights various aspects of the concept
of multivariate regular variation which was introduced in Chapter 1.
Recall from Theorem 4.2 in the previous chapter that any elliptically
distributed random vector has a stochastic representation of the form
(see Theorem 4.2 for details)
X = fi + RAV, (5.1)
where the nonnegative random variable R and the random vector U are
independent. This simple structure of elliptical distributions enables
explicit computations of interesting quantities such as the coefficients
of tail dependence (see Definition 5.1 below) and spectral measures as¬
sociated with regularly varying random vectors (see Theorem 1.15).
This chapter is organized as follows. In Section 5.1 we prove that for
an elliptically distributed random vector X with representation (5.1),X is (multivariate) regularly varying with index a > 0 if and only if
R is regularly varying with index a > 0. We also show that if R is
125
126 Chapter 5. Extremes for elliptical distributions
regularly varying with index a > 0 and Qij > — 1 (see Definition 4.7),then (Xi, Xj) has tail dependence (see Definition 5.1 below) and we de¬
rive an expression for the coefficient of tail dependence from which we
conclude that the coefficient of tail dependence is fully determined by
the corresponding linear correlation coefficient (as defined in Definition
4.7) and the tail index of the radial random variable R in the general
representation. In Section 5.2 we explicitly compute the spectral mea¬
sure associated with a regularly varying elliptically distributed random
vector with respect to the usual Euclidean norm and the max-norm,
respectively. We find that the spectral measure depends only on the
choice of norm, the tail index a and the dispersion matrix S. Moreover,
the explicit expressions for the spectral measures allow us to interpret
the effect of the choice of norm on the spectral measure but also to
illustrate the multivariate tail behavior of regularly varying elliptically
distributed random vectors.
5.1 The connection between regular varia¬
tion and tail dependence
Perhaps the most commonly encountered measure of dependence of bi¬
variate extremes is the coefficient of upper (lower) tail dependence.
Recall that for a univariate distribution function F we denote by F_1
its (left-continuous) generalized inverse given by F~l(u) = inf{x G M :
F(x) > u] with the convention inf 0 = oo.
Definition 5.1 Let (Xi,X2)T be a random vector with marginal dis¬
tribution functions F\ and F2. The coefficient of upper tail dependence
of (Xi,X2)T is defined as
\u(Xi,X2) 4 limP(X2 > Fï\u) \ Xx > F^(u)),u-fl
provided that the limit \u(Xi,X2) G [0,1] exists. The coefficient of
lower tail dependence is defined as
Xl(Xi,X2) 4 limP(X2 < F~l(u) | Xi < F^(u)),
5.1. Regular variation and tail dependence 127
provided that the limit Xl(Xi,X2) G [0,1] exists. If \u(Xi,X2) > 0
(Xl(Xi,X2) > 0), then we say that (Xi,X2)T has upper (lower) tail
dependence.
Elliptically distributed random vectors are radially symmetric. Hence,
if (Xi,X2)T is elliptically distributed, then Xu(Xi,X2) = Xl(Xi,X2).If XU(X1,X2) = XL(Xi,X2) > 0, then we say that (Xi,X2)T has tail
dependence. See e.g. Nelsen [30] for details on radial symmetry. Note
that the tail dependence coefficients need not exist, see e.g. Example
1.24.
For a pair of random variables, upper (lower) tail dependence is a mea¬
sure of joint extremes. That is, it measures the probability that one
component is extremely large (small) given that the other one is ex¬
tremely large (small), relative to the marginal distributions. We have
introduced two concepts for measuring dependence of multivariate ex¬
tremes of random vectors, the coefficient of tail dependence (Definition
5.1) and the spectral measure associated with a regularly varying ran¬
dom vector (Definition 1.17). In the next theorem we clarify the connec¬
tion between these two concepts. We also derive an explicit expression
for the coefficient of tail dependence for two random variables with a
joint elliptical distribution.
Theorem 5.2 Let X = /i + RAXJ ~ Ed(fi,Z,(p), with S^ > 0 for
i = 1,..., d, and where fi, R, A and XJ are as in Theorem 4-2. Then
the following statements hold.
(i) R is regularly varying with index a > 0 if and only ifX is regularly
varying with index a > 0.
(ii) If R is regularly varying with index a; > 0, then
-tt/2
XU(X1,X3) = XL(X,,X3) =J{v/2-"eii)/*~" "". (5.2)f
'cosatdt
J (n/2—arcsin gij)/2
f*/2 cos« tdt
Remark 5.3 By applying Theorem 2.1 in Bingham and Inoue [8] one
can show that if £^ > 0, T,jj > 0, \gij\ < 1 and Xu(Xi,Xj) > 0 (or
128 Chapter 5. Extremes for elliptical distributions
equivalently Xl(X1,X3) > 0), then R is regularly varying with index
\ogF(R>u)a = — lim sup
u^oo logW
if, for some e > 0 and some a G (0,1), a is the only zero of
r1 uan
r uz,
r uan
r1 uz.du / du — / du / du
'o Vi — u2 Jo y/1 — u2 Jo \/l — u2 Jo y/ï u2
in {w G C : Ke(w) G (—e + a, e + a)}. Computations (using the software
package Maple) for a large number of a indicate that this is likely to
hold (i.e. that Xu(Xl,X3) > 0 implies that R is regularly varying).
Remark 5.4 Note that the tail dependence coefficient (5.2) is increas¬
ing in gl3 and decreasing in a. Also note that
C/2 cos" tdt.. J(tt/2—arcsin ol7)/2 _
hm ——
tt—— = 0.a^°° f*' cos« tdt
Remark 5.5 Let X ~ Ed(fi,T,,(p) with Xz ~ F% and X3 ~ F3. Note
that if limw|i F~ (u) < oo, i.e. if X% is a bounded random variable,
then there exists a uq G (0,1) such that the events {Xt > F~x(u)} and
{X3 > F~l(u)} are disjoint for u > uq, and hence
F(Xl>F-1(u),X3>F-1(u))hm i
= 0,«ti F(Xl>F~1(u))
i.e.Xu(Xl,X3) = XL(Xl,X3) = 0.
From the theorem above we can conclude that the bivariate marginals
of an elliptically distributed vector X have tail dependence if the radial
random variable R in the representation X = fi + RAXJ is regularly
varying with index a > 0. The linear correlation coefficient gl3 only ef¬
fects the magnitude of the coefficient of tail dependence. An interesting
consequence is that if X ~ Ed(fi, S, (p), then (Xt, X3)T can have a coef¬
ficient of tail dependence significantly larger than zero even if the linear
5.1. Regular variation and tail dependence 129
correlation coefficient for (Xi,Xj)T is zero or negative. In Figure 5.1
we have plotted the coefficient of tail dependence for a regularly vary¬
ing elliptically distributed bivariate random vector with uncorrelated
components as a function of the tail index a.
Proof of Theorem 5.2. (i) By Corollary 1.33 we can without loss
of generality assume that fi = 0. If rank(S) = k < d, denote by£(_1) A (A(-1))tA(-1) the generalized inverse of £, where A^-1) =
(ATA)~1AT, i.e. A(_1) solves A^~^A = Ik, where Ik denotes the k x k
identity matrix. Note that £(_1) = S_1 if rank(E) = d. By choosing
the norm |x|E = (xtTj^~1>x)1'2 in the definition of regular variation (by
Corollary 1.20 we are allowed to choose any norm), we obtain
P(|X|E > tx, X/|X|S G • ) ¥{R > tx, AXJ e )
|X|S > t) ¥(R > t)
F(R > tx) F(AU G • )
¥(R > t)
,—aIf R is regularly varying, then lim^oo F(R > tx)/F(R > t) = x~
and hence P(|X|S > ta,X/|X|E G • )/P(|X|E > t) A- x~aF(AV G • )as t ^ oo. Conversely, if P(|X|E > ta,X/|X|E G • )/P(|X|E > t) -^
x-a p^Q ^ . ^ as ^ _^ 00^ then we must have O = AXJ and lim^oo F(R >
tx)/F(R>t) =x~a.
(ii) For Qij e {—1,1} the conclusion follows immediately. Therefore we
only consider the case \gij\ < 1. First note that if Xu (Xi — fii, Xj —
fij )
exists, then Xjj(Xi — fii,Xj —fij) = Xu(Xi:Xj). Hence, we can without
loss of generality take fi = 0. Note that if X ~ Ed(0, S, (p), with Xi ~ Fi
and Xj ~ Fj, then F~l(u) = x/Ë~fË~F~1(u) for u G (0,1). Note also
that if limw|iF~1(u) < oo, i.e. if Xi is a bounded random variable,
then there exists a uq G (0,1) such that the events {Xi > F~x(u)} and
{Xj > F~l(u)} are disjoint for u > uq, and hence
F(Xl>F-1(u),X3>F-1(u))hm i
= 0.«ti F(Xi>Fl-1(u))
Hence, if Xu(Xi,X3) G (0,1] exists, then
F(Xi>^fY~iZ,Xj > y/Ë^z)Xv(Xi,Xj) = lim
F(Xi > ^JT~iz)
130 Chapter 5. Extremes for elliptical distributions
Since X = RAXJ,
'Xi \ d ( y/^a 0\ ( cos f
R
T
\xjJ \ y/ZjjQij y/^jjyi-ßij ) \sin^
where if ~ U(—tt,tt), i.e.
(Xi,Xj)T = (y/Ë~iR cos (f, x/^J](gijR cos if + yj1-g2jR sin if))
= (y/^uRcosif, \/ÊjjRsin(aicsingij + (f))T.
Hence, if Xjj(Xi,X3) G (0,1] exists, then
, , F(R cos (f > z,Rsin(axcsin gij + if) > z)XjjiXi. Xj)
= hm ——
r .UK *' JJz^oo F(Rcosf>z)
The numerator can be written as
F(Rcos if > z, R sin(arcsin gi3 + if) > z)1 1
= F(R> zmax(,
-—
: r),vcos if sin (arcsin g^ + if)
cos if > 0, sin(arcsin g^ + ip) > 0)
= - / P(i2> z/cost)dt,** J (tt/2—arcsin gij)/2
and the denominator can be written as
P(Ä cos 99 > z) = F(R > z/ cos if, cos 99 > 0)
1 W2
= - / F(R> z/cost)dt.
Suppose there is an a > 0 such that for every x > 0
¥(R > zx)j F(R > z) -r> x~a as z -+ 00, (5.3)
i.e. suppose Ä is regularly varying with a > 0. By Theorem 1.5.2 p. 22
in Bingham, Goldie and Teugels [7], for every 0 < a < b < 00,
¥(R > zx)j F(R > z) -r> x~a as 2 - 00,
5.2. Explicit computations of spectral measures 131
uniformly in x on [a, b]. In particular, with x = 1/ cost,
F(R> z/cost)/F(R> z) ^cosat as z -)> oo,
uniformly in £ on [0,7r/2 — e] for every e G (0,7r/2) Hence, for every
e G (0, tt/2) and a G [0, tt/2 - e],
r/2F(R> z/cost)n
r/2~ea ,
hmsup /m/n
'—-—-dt < / cosatdt + e
z^JJa F(R>Z) -Ja
n
r/2F(R>z/cost)n
r/2~e„ ,
liminf/v
, „
'
,
'dt > / cos" Mt.
p ä > z-
A0—»-OO I
If we let
C'%.
./0¥(R> z/cost)dtw \ A J(7r/2-arcsmgi3)/2 V / /
f*/2F(R> z/cost)dt
then
f^/2-6 cos« tdtJ(ff/2-arcsinglJ)/2
< liminfA(z)J^7 ecos«tdt + e
z^°°
< lim sup A(z)z—>oo
<
r/2~e cos" tdt + eJ (tt/2—arcsin gl3 )/2
S;/2~e cos« Mt
Letting e —>• 0 yields
f71-/2 cos« ^\ /v V \
—
J(^/2-arcsineîJ)/2
/0X cos« tdt
Moreover, because elliptically distributed random vectors are radially
symmetric about fi, X\j(Xi,X3) = Xi,(Xi,X3). D
5.2 Interpretations and explicit computa¬
tions of spectral measures
In this section we discuss how to interpret the spectral measure with
respect to different norms. The discussion is general but in the case of
132 Chapter 5. Extremes for elliptical distributions
elliptical distributions we can explicitly compute the spectral measure
with respect to different norms and compare different choices. Real
data, e.g. financial asset price log returns, often indicate that the un¬
derlying distribution is elliptical or at least close to elliptical, and many
statistical models are based on the assumption of ellipticality. Hence
the following discussion should be relevant for many applications, espe¬
cially in risk management. See Breymann, Dias and Embrechts [10] for
an interesting empirical study of dependence and extremes for bivariate
time series of foreign exchange data.
By Corollary 1.20 we know that if a random vector X is regularly varying
with respect to some norm on IRd, then it is regularly varying with
respect to every norm on FLd. For every choice of the norm the spectral
measure is a measure of dependence between extreme values. However,
the choice of norm becomes essential when interpreting the spectral
measure. The choice of norm must be related to the question we are
trying to answer. A natural question would be: What is the dependence
between the components of a random vector given that at least one of
its components is extreme? In the literature (see e.g. Stäricä [39]) most
authors consider the Euclidean 2-norm, | • |2- However, if we want a
measure of dependence between the components - given that at least
one of the components is extreme - then we should use the max-norm
IXloo = max{|Xi|,..., |-Xd|}. Clearly, if we take x = 1 in equation (1.2)of Theorem 1.15, we have that
P(eoo G • ) 4 hm P(X/|X|00 G • | ixi«, > t)t—>-oo
= hmP(X/|X|00G-md=1{|Xj|>t})
from which it is seen that the max-norm corresponds to the question
posed. However, if the components are not identically distributed,
then it might be more natural to condition on the event Uj=1{|Xj| >
G~l(u)}, where Gj is the distribution function of \Xj\ and u t 1- For
X ~ Ed(0, S, <p) this is achieved by considering the weighted max-norm
Ix|oo,e = max{|Xi|/v/Ëïï, •••> \xd\/y/^dd}, since in this case,
P(eoo,E G • ) = lim P(X/|X|00;E G • I ixi^e > t)
= limP(X/|X|00;E G • | Ud=i{\X3\ > G-\u)}),
5.2. Explicit computations of spectral measures 133
is the spectral measure of X with respect to the norm | • |oojE. The
corresponding question in this case would be: What is the dependence
between the components of a random vector given that at least one of
its components is extreme relative to its marginal distribution?
In the following two examples we compute the spectral measure with
respect to the Euclidean 2-norm and the max-norm for bivariate regu¬
larly varying elliptical distributions. This can also be done for elliptical
distributions of higher dimension, but the corresponding computations
in spherical coordinates become quite tedious.
Example 5.6 Let X ~ E2(0,T,,(p), with En, £22 > 0> t>e regularly
varying with index a > 0, and let X = RAXJ be a stochastic repre¬
sentation according to Theorem 4.2. Without loss of generality we can
choose A and U such that
/Xi\ ±R( V^Ti 0 \ /cosyA
\X2) y yf^22g\2 y/T^/i^/l - q\2 J \simf)'
where p ~ U(—ir/2, Sir/2), i.e.
(Xi,X2)T = (y/ËïÏRcosip, yfË2~2~(gi2Rcosif + y 1 - g22Rsimf))T
(y/EiiRcosif, y/^22Rsin(arcsing12 + f))T
Let
f(t) = (Sn cos2 t + E22 sin2(arcsin gi3 +t)) ,
f -tt/2, t = -7r/2,
g(t) 4 J arctan (^§^12 + V1 ~ &tant)) ,t G (-tt/2, tt/2),
{ g(t-7r) + 7T, te [7r/2,37r/2).
Then,
RAv=R\Avhj§r±Rfacosf):\AXJ\2 \smg(if)/
Since X is regularly varying and X/|X|2 has continuous distribution on
S2, there exists a random vector O such that for every x > 0 and every
S B(S£),
lim m\AVh>rAV/\AVh e 5)=
z^oo ¥(R\AXJ\2 > z)K '
134 Chapter 5. Extremes for elliptical distributions
Moreover, by Theorem 5.2, R is regularly varying which implies that
there exists a slowly varying function L such that F(R > x) = x~aL(x).Let Sgljg2 = {(cost, sint)T : t G (0i,02)}, where by symmetry we can
assume that —tt/2 < 9i < 92 < tt/2. The case \gi2\ = 1 is trivial, so we
consider only the case |^»i2| < 1. Then, for t G (—tt/2, tt/2),
q l(t) = arctan —. ( .tant — gi2
and
F(R\AXJ\2 > zx,AXJ/\AXJ\2 e S)lim
F(R\AXJ\2 > z)
,
J£h$ z-«x-af(trL(zx/f(t))dt= hm ——l-t5
J^z-"f(t)"L(z/f(t))dt
« ,
i;-^f(t)aL(zx/f(t))/L(z)dt—
x hm—-—^—^
z-°° Cf(t)«L(z/f(t))/L(z)dt
J9-u^ f(t)adt
/o /Wad*
-a//-H^O (Sl1 C0S2 t + S22 sin2(arCSm Ö12 + ^))"/2 dt
X
Jo^
(Sn cos21 + £22 sin2(arcsin £12 + t)) dt
The third equality follows from the fact that L(zx)/L(z) —> 1 uniformly
in x on intervals [a, b], 0 < a < b < 00 (Theorem 1.5.2 p. 22 in Bingham,
Goldie and Teugels [7]) and from the fact that there exist constants
0 < ci < c2 < 00 such that c\ < l/f(t) < c2 for all t G [-tt/2,tt/2].Now we can identify the spectral measure as
In-i(fti (Sn cos21 + S22 sin2(arcsin g12 + t))a/2 dt
p(e g s9ue2) =g 2nl} ^72—•JQn (Sn cos21 + £22sin2(arcsin^>i2 + t))" dt
Note that the spectral measure depends on the tail index a. Further¬
more, note that lima^0P(O e S) = F(AXJ/\AXJ\2 e S) for all S G
B(S>2_1). We see that the spectral measure is absolutely continuous and
hence it has a density. The density is plotted in Figure 5.2 for bivariate
5.2. Explicit computations of spectral measures 135
regularly varying elliptical distributions with (£n, £22, Q12) = (1,1,0.5)and with tail indices a = 0, 2,4, 8,16 (we write a = 0 for the limit mea¬
sure lima_>oP(0 G • )). From this figure it can be seen that as a
increases - that is as the tails become lighter - the probability mass
becomes more concentrated in the main directions of the ellipse (in this
case 7r/4 and 57r/4). Note also that the density can be explicitly com¬
puted by differentiating expression (5.4).
Example 5.7 Let us now compute the spectral measure with respect
to the norm | • |oo- Proceeding analogously to the previous example but
replacing the function / by
f(t) = max{-\/£ii I cost|, \/£22 | sin(arcsingi2 + t)\},
we find thatAXJ
RAV = RlAXJlov——|-AU loo
where AU/jAU^ = f(if) with f ~ U(-tt/2, 3tt/2). Following the
computations in the previous example we obtain the spectral measure
with respect to the max-norm as
P(© G Sdue2)
//-i(01) (max{^/£77 I cost|, y/T^ | sin(arcsin gX2 + t) | })a dt
JQn (max{\/£ii I cost|, a/^22 | sin(arcsin gi2 +t)\})a dt
where Sq1}q2 is the radial projection of Sq1iq2 (see Example 5.6) on S^,the unit circle with respect to the max-norm. The density of the spectral
measure is plotted in Figure 5.3 for bivariate regularly varying ellipti¬
cal distributions with (£n, £22, £12) = (1,1,0.5) and with tail indices
a = 0, 2,4, 8,16. From this figure it can be seen that as a increases
- that is, as the tails become lighter - the probability mass becomes
less concentrated in the main directions of the ellipse (in this case 7r/4and 57r/4). This is quite intuitive, for (bivariate) regularly varying el¬
liptical distributions with lighter tails, the probability of joint extremes
(that both components are extreme) becomes very small compared to
the probability that one component is extreme. This can be seen from
the fact that the coefficient of tail dependence tends to zero as the tail
index increases (see Remark 5.4 and Figure 5.1).
136 Chapter 5. Extremes for elliptical distributions
0.5-
0.4-
0.3-
0.2-
0.1
_- , , , , i , , , , i , , , , i , , , , i , , , , i ,
0 2 4 6 8 10
Figure 5.1: The coefficients of upper and lower tail depen¬
dence for regularly varying bivariate elliptical distributions with
(En, E22, 012) = (1,1, 0), as a function of the tail index a (see Remark
5.4).
Note the striking difference between the spectral measure with respect to
the Euclidean norm and the spectral measure with respect to the max-
norm. By choosing a norm which does not correspond to the question
one is trying to answer, one might draw completely wrong conclusions
about dependences between extremes. The best illustration of this is
the comparison of Figure 5.2 with Figure 5.3.
5.2. Explicit computations of spectral measures 137
! 'n ; ',; i i tit < i,
Figure 5.2: Densities of the spectral measure ofX ~ E2(fi,Yj,(p) with
respect to the Euclidean 2-norm, where (En, £22, 012) = (1,1, 0.5), and
ta«Z index a = 0, 2,4, 8,16. Larger tail indices correspond to higher peaks
(see Example 5.6).
Figure 5.3: Densities of the spectral measure ofX ~ E2(fi,Tj,(p) with
respect to the max-norm, where (En, £22,012) = (1,1,0.5), and tail
index a = 0,2,4, 8,16. Larger tail indices correspond to higher peaks
(see Example 5.7).
Bibliography
[i
[2
[s:
[4:
[5
[9
Adler, R.J., Feldman, R.E. and Taqqu, M.S. (1998) A Prac¬
tical Guide to Heavy Tails: Statistical Techniques and Applications.
Birkhäuser, Boston.
Basrak, B. (2000) The Sample Autocorrelation Function of Non-
Linear Time Series. PhD Thesis. University of Groningen.
Basrak, B., Davis, R.A. and Mikosch, T. (2002a) Regular vari¬
ation of GARCH processes. Stochastic Process. Appl. 99, 95-115.
Basrak, B., Davis, R.A. and Mikosch, T. (2002b) A charac¬
terization of multivariate regular variation. Ann. Appl. Probab. 12,
908-920.
Billingsley, P. (1968) Convergence of Probability Measures, 1st
edition. Wiley, New York.
Billingsley, P. (1995) Probability and Measure, 3rd edition. Wi¬
ley, New York.
Bingham, N.H., Goldie, C.M. and Teugels, J.L. (1987) Reg¬
ular Variation. Cambridge University Press, Cambridge.
Bingham, N.H. and Inoue, A.Y. (2000) Tauberian and Mercerian
theorems for systems of kernels. J. Math. Anal. Appl. 252, 177-197.
Braverman, M., Mikosch, T. and Samorodnitsky, G. (2002)Tail probabilities of subadditive functionals acting on Levy pro¬
cesses. Ann. Appl. Probab. 12, 69-100.
139
140 Bibliography
[10] Breymann, W., Dias, A. and Embrechts, P. (2003) Depen¬
dence structures for multivariate high-frequency data in finance.
Quant. Finance 3(1), 1-16.
[11] Cambanis, S., Huang, S. and Simons, G. (1981) On the theory
of elliptically contoured distributions. J. Multivariate Anal. 11, 368-
385.
[12] Daley, D.J. and Vere-Jones, D. (1988) An Introduction to the
Theory of Point Processes. Springer-Verlag, New York.
[13] Davis, R.A. and Hsing, T. (1995) Point process and partial sum
convergence for weakly dependent random variables with infinite
variance. Ann. Probab. 23, 879-917.
[14] de Haan, L. and Lin, T. (2002) On convergence toward an ex¬
treme value limit in C[0,1]. Ann. Probab. 29, 467-483.
[15] Embrechts, P., Goldie, CM. and Veraverbeke, N. (1979)
Subexponentiality and infinite divisibility. Z. Wahrsch. verw. Gebi¬
ete 49, 335-347.
[16] Embrechts, P., McNeil A. and Straumann D. (2002) Cor-
relation and dependence in Risk Management: properties and pit¬
falls. In: Risk Management: Value at Risk and Beyond, ed. M.A.H.
Dempster. Cambridge University Press, Cambridge, pp. 176-223.
[17] Fang, K.-T., Kotz, S. and Ng, K.-W. (1987) Symmetrie Mul¬
tivariate and Related Distributions. Chapman & Hall, London.
[18] GiNÉ, E., Hahn, M.G. and Vatan, P. (1990) Max-infinitely di¬
visible and max-stable sample continuous processes. Probab. Theor.
Rel. Fields 87, 139-165.
[19] Gradshteyn, LS. and Ryzhik, I.M. (2000) Table of Integrals,
Series, and Products, 6th edition. Academic Press, Orlando.
[20] Gihman, LI. and Skorohod, A.V. (1974) The Theory of
Stochastic Processes I. Springer-Verlag, Berlin.
[21] Gihman, LI. and Skorohod, A.V. (1975) The Theory of
Stochastic Processes II. Springer-Verlag, Berlin.
Bibliography 141
[22] Hult, H. and Lindskog, F. (2002) Multivariate extremes, ag¬
gregation and dependence in elliptical distributions. Adv. in Appl.
Probab. 34, 587-608.
[23] Hult, H. and Lindskog, F. (2002) Multivariate regular variation
for additive processes. Submitted.
[24] Hult, H. and Lindskog, F. (2003) On regular variation for
stochastic processes. Submitted.
[25] Joe, H. (1997) Multivariate Models and Dependence Concepts,
Chapman & Hall, London.
[26] Kallenberg, O. (1983) Random Measures, 3rd edition.
Akademie-Verlag, Berlin.
[27] Kesten, H. (1973) Random difference equations and renewal the¬
ory for products of random matrices. Acta Math. 131, 207-248.
[28] Lindskog, F., McNeil, A.J. and Schmock, U. (2003) A
note on Kendall's tau for elliptical distributions. In: Credit
Risk. Measurement, Evolution and Management, eds. G. Bol, G.
Nakhaeizadeh, S.T. Rachev, T. Ridder, K.-H. Vollmer. Physica-
Verlag, A Springer-Verlag Company, Heidelberg, pp. 149-156. Avail¬
able at www.risklab.ch/Papers
[29] Mikosch, T. (2003) Modeling dependence and tails of financial
time series. To appear in: Extreme Values in Finance, Telecomuni-
cations and the Environment, Chapman & Hall, London.
[30] Nelsen, R. (1999) An Introduction to Copulas. Springer-Verlag,
New York.
[31] Pratt, J. (1960) On interchanging limits and integrals. Ann.
Math. Statist. 31, 74-77.
[32] Rachev, S.T. (2003) Handbook of Heavy-Tailed Distributions in
Finance. Elsevire, Amsterdam.
[33] Resnick, S.I. (1986) Point processes, regular variation and weak
convergence. Adv. in Appl. Probab. 18, 66-138.
142 Bibliography
[34] Resnick, S.I. (1987) Extreme Values, Regular Variation, and
Point Processes. Springer-Verlag, New York.
[35] Resnick, S.I. (2002) On the foundations of multivariate
heavy tail analysis. Cornell Report no. 1335, available at
www. orie. Cornell, edu/
[36] Rosinski, J. and Samorodnitsky, G. (1993) Distributions of
subadditive functionals of sample paths of infinitely divisible pro¬
cesses. Ann. Probab. 21, 996-1014.
[37] Rvaceva, E.L. (1962) On the domains of attraction of multidi¬
mensional distributions. In: Selected Translations Math. Stat. Prob.,
vol. 2, pp. 183-207. Inst. Math. Statistics-Amer. Math. Soc.
[38] Sato, K.-I. (1999) Levy Processes and Infinitely Divisible Distri¬
butions. Cambridge University Press, Cambridge.
[39] StÄricÄ, C. (1999) Multivariate extremes for models with con¬
stant conditional correlations. Journal of Empirical Finance 6, 515-
553.
[40] Willekens, E. (1987) On the supremum of an infinitely divisible
process. Stochastic Process. Appl. 26, 173-175.
Curriculum Vitae
Personal Data
Name:
Date of Birth:
Nationality:
Education
2001 - date:
1994 - 2000:
1991 - 1994:
Carl Filip Lindskog
20.10.1975
Swedish
Ph.D. student in mathematics at ETH Zürich;
Supervisor: Prof. Dr. Paul Embrechts
Studies in mathematics and physics at the
MSc programme in Engineering Physics,
Royal Institute of Technology (KTH), Stock¬
holm;
MSc awarded in February 2000
Bromma Gymnasium, Stockholm
Employment
2000 - date: Researcher at RiskLab, ETH Zürich
Research Interests
2000 - date: Stochastic processes, multivariate extreme
value theory, dependence modelling, Risk
Management, Mathematical Finance
143