Journal of the Korean Statistical Society 41 (2012) 555–562

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Journal of the Korean Statistical Society

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On some dependence structures for multidimensional Lévy drivenmoving averagesShibin Zhang ∗Department of Mathematics, Shanghai Maritime University, 1550 Haigang Avenue In New Harbor City, Shanghai 201306, China

a r t i c l e i n f o

Article history:Received 18 October 2011Accepted 20 January 2012Available online 17 February 2012

AMS 2000 subject classifications:primary 60G99secondary 62H99

Keywords:Lévy processLévy copulaMoving averageInfinitely divisible distributionOrnstein–Uhlenbeck

a b s t r a c t

The Lévy copula can describe the dependence structure of a multidimensional Lévyprocess or a multivariate infinitely divisible random variable. Suppose the Lévy copula of amultidimensional Lévy process is known. We present the Lévy copula of the Lévy measureof the moving average driven by the multidimensional Lévy process. If there exist somespecial dependence structures among the components of the Lévy process, we give somedependence invariance properties after the transform of the moving average.

© 2012 The Korean Statistical Society. Published by Elsevier B.V. All rights reserved.

1. Introduction

The concept of copulas is used to describe the dependence structures of multivariate probability distributions. As ananalogue to distributional copulas, the concept of Lévy copulas was introduced by Tankov (unpublished) for Lévy measureson Rd

+and a generalization to Lévy measures on Rd can be seen in Cont and Tankov (2004) and Kallsen and Tankov (2006).

The Lévy copula can describe the dependence among components and does not depend on their individual laws. For amultidimensional Lévy process, the dependence of its components can be uniquely determined by its Lévy copula (see e.g.Kallsen and Tankov (2006)). Recently, Lévy driven moving averages (MAs) have drawn considerable attention in theoriesand applications (see e.g. Barndorff-Nielsen and Shephard (2001), Basse and Pedersen (2009) and Marquardt (2006)). Ifthe driving process is a Lévy process, then the law of the MA is infinitely divisible (see Lemma 1). Since the dependenceof components of a multivariate infinitely divisible random variable or a multidimensional Lévy process is completelycharacterized by the Lévy copula, this paper will focus on the Lévy copula connections between the driving Lévy processand the law of the driven MA. These connections would be useful, for example, to construct a multidimensional Lévy drivenMA with a specified dependent structure. Lévy driven Ornstein–Uhlenbeck (O–U) processes are an important special caseof Markov processes with jumps. The transition law and the stationary law play crucial roles in describing the properties ofthe O–U process, which can be treated as the law of a Lévy driven average (see Example 2).

In Section 2, we recall some related definitions and symbols of Lévy processes and Lévy copulas. In Section 3, for the MAdriven by a multidimensional Lévy process, we present our general results about the Lévy copula connections between thedriving Lévy process and the marginal law of the driven process. In Section 4, for some special dependence structures ofcomponents of the driving Lévy process, we provide some dependence invariance properties after the transform of the MA.

∗ Tel.: +86 21 38282230; fax: +86 21 38282209.E-mail address: sbzhang@shmtu.edu.cn.

1226-3192/$ – see front matter© 2012 The Korean Statistical Society. Published by Elsevier B.V. All rights reserved.doi:10.1016/j.jkss.2012.01.004

556 S. Zhang / Journal of the Korean Statistical Society 41 (2012) 555–562

2. Preliminaries

In this section, we shall recall a few facts on Lévy processes and Lévy measures, and some definitions related to the Lévycopula; for further details, the reader is referred to Barndorff-Nielsen and Lindner (2007), Cont and Tankov (2004), Kallsenand Tankov (2006), Sato (1999) and Tankov (unpublished).

Let L = L(t), t ≥ 0 be a Lévy processwith values inRd defined on a filtered probability space (Ω, (Ft)t≥0, F , P), wherewe assume that it has cádlág sample paths. For each t > 0, the random variable L(t) has an infinitely divisible distribution,whose characteristic function has a Lévy–Khintchine representation

E[ei⟨z,L(t)⟩] = exp

t

i⟨γ , z⟩ −

12zTAz +

Rd

ei⟨z,x⟩ − 1− i⟨z, x⟩1|x|≤1

ρ(dx)

, z ∈ Rd, (1)

where ⟨·, ·⟩ denotes the inner product in Rd, |x| =√⟨x, x⟩ denotes the Euclidean norm of x = (x1, . . . , xd), γ ∈ Rd

and A is a symmetric nonnegative definite d × d matrix. The Lévy measure ρ is a measure on Rd satisfying ρ(0) = 0and

Rd\0 1 ∧ |x|

2ρ(dx) < ∞. Throughout this paper, we shall work with a two-sided Lévy process L = L(t), t ∈ Rconstructed by taking two independent copies L1(t), t ≥ 0, L2(t), t ≥ 0 of a one-sided Lévy process and settingL(t) = L1(t)1t≥0 + L2(−t−)1t<0.

In the sequel, we shall need a special interval associated with any x ∈ R

I(x) :=(x,∞), x ≥ 0,(−∞, x), x < 0.

For every Lévy measure ρ on Rd, the tail integral W = Wρ can be defined as the functionW : Rd→ R given by

W (x1, . . . , xd) :=d

i=1

sgn(xi)ρI(x1)× · · · × I(xd)

(2)

for any (x1, . . . , xd) ∈ Rd. Here and hereafter, R = (−∞,∞] and sgn(x) = 1x≥0 − 1x<0.Let I ⊂ 1, . . . , d be non-empty. The I-marginal tail integral W I of ρ is given by

W I(xi)i∈I := lima→0−

(xi)i∈Ic ∈a,0|I

c |

Wx1, . . . , xd

i∈Ic

sgn(xi)

for any (xi)i∈I ∈ R|I|, where |I| denotes the cardinality of the set I and Ic := 1, . . . , d \ I . In fact, W I is the tail integral ofthe I-marginal Lévy measure of ρ, which is defined by

ρ I(A) := ρx ∈ Rd

: (xi)i∈I ∈ A, A ⊂ R|I| \ 0. (3)

Correspondingly, W I is related to ρ I by W I(xi)i∈I

=|I|

i=1 sgn(xi)×i∈I I(xi)

ρ I(duI) for (xi)i∈I ∈ R|I| \ 0. To simplifynotation, we denote one-dimensional margins by Wi := W i, i.e., Wi(xi) = W (0, . . . , 0, xi, 0, . . . , 0) for any xi ∈ R, i ∈1, . . . , d.

For a, b ∈ Rd wewrite a ≤ b if ak ≤ bk, k = 1, . . . , d. In this case, let (a, b] denote a right-closed left-open interval of Rd,i.e., (a, b] = (a1, b1] × · · · × (ad, bd]. A function C : Rd

→ R is called d-increasing if

c∈a1,b1×···×ad,bd(−1)N(c)C(c) ≥ 0

for all a, b ∈ Rd with a ≤ b, where N(c) := #k : ck = ak, k = 1, . . . , dwith c = (c1, . . . , cd).Let C : Rd

→ R be a d-increasing function such that C(u1, . . . , ud) = 0 if ui = 0 for at least one i ∈ 1, . . . , d. For anynon-empty index set I ⊂ 1, . . . , d, the I-margin of C is the function C I

: R|I| → R, defined by

C I((ui)i∈I) := lima→∞

(ui)i∈Ic ∈−a,∞|I

c |

Cu1, . . . , cd

i∈Ic

sgn(ui)

for any (ui)i∈I ∈ R|I|.Now, we restate Definition 3.1 of Kallsen and Tankov (2006) as follows.

Definition 1. A function C : Rd→ R is called a Lévy copula if

(i) C(u1, . . . , ud) = ∞ for (u1, . . . , ud) = (∞, . . . ,∞),(ii) C(u1, . . . , ud) = 0 if ui = 0 for at least one i ∈ 1, . . . , d,(iii) C is d-increasing, and(iv) Ci(u) = u for any i ∈ 1, . . . , d, u ∈ R.

S. Zhang / Journal of the Korean Statistical Society 41 (2012) 555–562 557

3. Lévy copula connections for multidimensional Lévy driven MAs

In this section, we shall consider the MA process X = X(t), t ∈ R in Rd, which is given by

X(t) =

Rf (t − s)L(ds), t ∈ R, (4)

where f : R→ R is ameasurable function and L = (L1, . . . , Ld) is a d-dimensional Lévy process.We say that f is L-integrableif there exists a sequence of simple functions ( fn)n≥1 such that fn → f almost surely with respect to the Lebesgue measureand limn→∞

A fn(s)L(ds) exists in probability for all A ∈ B(R). In this case we define

R f (s)L(ds) as the limit in probability

of

R fn(s)L(ds). Note that the integral is independent of the choice of approximating functions fn (see p. 460 of Rajput andRosinski (1989)).

The following lemma is a multidimensional version of Theorem 2.7 of Rajput and Rosinski (1989). When we generalizethe results of Theorem2.7 of Rajput and Rosinski (1989) to themultidimensional case,we find there is no essential differencein the proof.

Lemma 1. Let L be a d-dimensional Lévy process with characteristic function (1), and f : R→ R be a Borel measurable function.Then f is L-integrable if and only if the following three conditions hold:

(i)

R f (s)2 ds <∞ if A = 0,(ii)

R

Rd(|f (s)x|2 ∧ 1)ρ(dx)ds <∞, and

(iii)

R

f (s)γ + Rd x(1|f (s)x|≤1 − 1|x|≤1)ρ(dx)ds <∞.

Further, if f is L-integrable, then integral (4) is infinitely divisible with the characteristic function

E[ei⟨z,X(t)⟩] = exp

i⟨γf , z⟩ −

12zTAf z +

Rd

ei⟨z,ξ⟩ − 1− i⟨z, ξ⟩1|ξ |≤1

νf (dξ)

,

where γf , Af and νf are given by

γf =

Rf (s)

γ +

Rd

x(1|f (s)x|≤1 − 1|x|≤1)ρ(dx)

ds, Af = A

Rf (s)2 ds

and

νf (B) =

Rdρ(dy)

R1Bf (s)y

ds, B ∈ B(Rd). (5)

Remark 1. (1) Actually, if the driving Lévy process has non-Brownianmotion part, f is L-integrable if and only if (ii) and (iii)of Lemma 1 hold. (2) If f is L-integrable, then f is Li-integrable for all i ∈ 1, . . . , d.

In contrast to the Gaussian parts of L and X , the dependence structure of the components of the non-Gaussian partsis complicatedly covered by the ordinary copula. Therefore, in the sequel, this paper will only consider the dependencestructure of the components for the non-Gaussian parts of L and X , which are completely determined by the Lévy measuresρ and νf , respectively. To obtain our main results, we still need the following condition.

Condition 1. Suppose f (s) ≥ 0 for all s ∈ R and suppf = ∅, where suppf is the support of f . Further, f , γ and ρsatisfy (ii)–(iii) of Lemma 1.

LetW (x1, . . . , xd) =d

i=1 sgn(xi)

I(x1)×···×I(xd)ρ(dy) for (x1, . . . , xd) ∈ Rd

\ 0, i.e.,W (x1, . . . , xd) is the tail integral ofthe Lévy measure of L. Let Uf (x1, . . . , xd) be the tail integral of the Lévy measure of the MA process X . Then, we obtain thefollowing relationships.

Lemma 2. Under Condition 1, we have for x ∈ Rd,

Uf (x) =

R\f−1(0)W ((1/f (s))x) ds (6)

and for (xi)i∈I ∈ R|I|,

U If

(xi)i∈I

=

R\f−1(0)

W I((1/f (s))xi)i∈I ds, (7)

where f −1(0) = s : s ∈ R, f (s) = 0.

558 S. Zhang / Journal of the Korean Statistical Society 41 (2012) 555–562

Proof. By the definition of tail integral (2), we have for x ∈ Rd\ 0,

Uf (x) =d

i=1

sgn(xi)νfI(x1)× · · · × I(xd)

=

di=1

sgn(xi)

Rdρ(dy)

R\f−1(0)

1I(x1)×···×I(xd)f (s)y

ds

=

di=1

sgn(xi)

R\f−1(0)ds

I((1/f (s))x1)×···×I((1/f (s))xd)ρ(dy) =

R\f−1(0)

W ((1/f (s))x) ds.

At x = 0, from the definition of tail integral (2), equality (6) still holds. This proves (6).We define X I and LI as X I(t) := (X i(t))i∈I and LI(t) := (Li(t))i∈I , respectively. From the integral form (4), it is obviously

true that for any non-empty set I ⊂ 1, . . . , d,

X I(t) =

Rf (t − s)LI(ds), t ∈ R.

According to the definition of I-marginal tail integral of the Lévy measure and equality (6), we have for (xi)i∈I ∈ R|I|,

U If

(xi)i∈I

= lim

a→0−

(xi)i∈Ic ∈a,0|I

c |

Ufx1, . . . , xd

i∈Ic

sgn(xi)

= lima→0−

(xi)i∈Ic ∈a,0|I

c |

R\f−1(0)

W(1/f (s))x1, . . . , (1/f (s))xd

dsi∈Ic

sgn(xi)

=

R\f−1(0)

W I((1/f (s))xi)i∈I ds.This proves (7).

If W (x) is the tail integral of the Lévy measure of a one-dimensional Lévy process, we define the inverse ofW (x) by

W←(w) =

infx > 0 : W (x) ≤ w, w ≥ 0,supx < 0 : W (x) ≥ w, w < 0.

Let L be a d-dimensional Lévy process and X be the MA driven by L. By Sklar’s theorem for Lévy copulas (e.g. Theorem 3.6of Kallsen and Tankov (2006)), there exists a unique Lévy copula on RanW1 × · · · × RanWd (where Ran denotes range of amapping and S denotes closure of a set S), denoted by C, such that the tail integrals of L satisfy

W I(xi)i∈I = C I

Wi(xi)i∈I

(8)

for any non-empty I ⊂ 1, . . . , d and any (xi)i∈I ∈ R|I|. Under Condition 1, there exists a unique Lévy copula onRanUf 1 × · · · × RanUfd, denoted by C, such that the tail integrals of X(t) satisfy

U If

(xi)i∈I

= C

I

Ufi(xi)i∈I

(9)

for any non-empty I ⊂ 1, . . . , d and any (xi)i∈I ∈ R|I|. The following theorem explains the relation between C and C.

Theorem 1. Let L be a d-dimensional Lévy process with Lévy copula C and X be the MA driven by L. Under Condition 1, we havefor any non-empty I ⊂ 1, . . . , d and any x ∈ Rd,

CI

Ufi(xi)i∈I

=

R\f−1(0)

C I

Wi(1/f (s))xi

i∈I

ds. (10)

Moreover, if all the one-dimensional marginal Lévy measures of L are infinitely divisible and have no atoms, then

CI

(ui)i∈I=

R\f−1(0)

C I

Wi(1/f (s))U←fi (ui)

i∈I

ds. (11)

Also, the expression

R\f−1(0) C IW1(1/f (s))U←f 1 (u1)

, . . . ,Wd

(1/f (s))U←fd (ud)

ds is a Lévy copula.

Proof. Since equalities (7), (8) and (9) hold, equality (10) holds obviously.

S. Zhang / Journal of the Korean Statistical Society 41 (2012) 555–562 559

By the definition of the marginal Lévy measure (3) and the Lévy measure expression (5) of the law of X(t), we have

νif (Bi) = νf

Ri−1× Bi × Rn−i

=

Rd

ρ(dy)

R1Bi

f (s)yi

ds

=

R

ρi(dyi)

R1Bi

f (s)yi

ds =

R\f−1(0)

ρi(1/f (s))Bi

ds, Bi ∈ B(R), (12)

where νf and ρ are the Lévy measures of the law of X(t) and the driving Lévy process L(t), respectively. From (12), wenote that if all the one-dimensional marginal Lévy measures of L are infinitely divisible and have no atoms, so are the one-dimensional marginal Lévy measures of the law of X(t). Thus, in (10), set xi = U←fi (ui), then Ufi

U←fi (ui)

= ui, this proves

(11).Since C is a Lévy copula and for each i ∈ 1, . . . , d the function U←fi is monotone decreasing on ui > 0 or on ui < 0, we

can check easily that

R\f−1(0) C IW1(1/f (s))U←f 1 (u1)

, . . . ,Wd

(1/f (s))U←fd (ud)

ds satisfy (i)–(iii) of Definition 1. Due to

(7), we obtain for any i ∈ 1, . . . , d,R\f−1(0)

Wi(1/f (s))U←fi (ui)

ds = Ufi

U←fi (ui)

= ui.

Now, it is easy to check (iv) of Definition 1 since C is a Lévy copula. This completes the proof.

Remark 2. If R \ f −1(0) is an unbounded set, then equality (11) holds only on the condition that all the one-dimensionalmarginal Lévy measures of L have no atoms.

Example 1. Suppose Li, i = 1, 2, . . . , d are d one-dimensional α-stable Lévy processes with respective tail integrals

Wi(xi) =cix−α

i , xi > 0,−ci|xi|−α, xi < 0, i = 1, . . . , d, (13)

where 0 < α < 2, and ci and ci are two positive constant. Assume further that L = (L1, . . . , Ld)T is a d-dimensional Lévyprocess with Lévy copula C and X is the MA driven by L. Then, according to (7), we have for any i ∈ 1, . . . , d,

Ufi(xi) =

ci

R\f−1(0)

( f (s))α ds

x−αi , xi > 0,

−ci

R\f−1(0)

( f (s))α ds

|xi|−α, xi < 0.

(14)

From (11), we obtain

CI

(ui)i∈I=

R\f−1(0)

C I

( f (s))α

R\f−1(0)( f (s))α ds

ui

i∈I

ds (15)

for any non-empty I ⊂ 1, . . . , d and any (ui)i∈I ∈ R|I|.

Example 2. Let µ > 0 be a parameter and L be a d-dimensional Lévy process. The process

X(t) = t

−∞

e−µ(t−s) L(ds), t ∈ R,

is called a Lévy driven Ornstein–Uhlenbeck (O–U) process, which is the MA with

f (s) = e−µs1[0,∞)(s), s ∈ R. (16)

According to Theorem 17.5 of Sato (1999), if the Lévy measure ρ of L satisfies|x|>2

log |x|ρ(dx) <∞, (17)

then the process X is well defined and has a self-decomposable stationary law. Note that for any fixed ∆ > 0,

X(t) = e−µ∆X(t −∆)+

t

t−∆

e−µ(t−s) L(ds).

560 S. Zhang / Journal of the Korean Statistical Society 41 (2012) 555–562

This implies that the O–U process X is a Markov process. In fact, it is a homogeneous Markov process (see Lemma 17.1 ofSato (1999)). For any fixed ∆ > 0, let Y (t) =

tt−∆

e−µ(t−s) L(ds). Then, the transition law of the process X is determined bythe distribution of Y (t), which can be treated as an MA with

f (s) = e−µs1[0,∆)(s), s ∈ R. (18)

Since both the transition law and the stationary law of the O–U process are determined by Lévy driven MAs, the Lévycopula connections between the driving Lévy process and the transition law and between the driving Lévy process and thestationary law can be treated as the special cases of Theorem 1.

4. Some dependence invariance properties after the transform of the MA

We begin with the independent case.We shall find that the independence of components is invariant after the transformof the MA.

Proposition 1. Let L1, . . . , Ld be components of a d-dimensional Lévy process L, and X be the MA driven by L. Under Condition 1,the one-dimensional marginal laws of X(t) are independent if and only if the components L1, . . . , Ld are independent.

Proof. Since for any i ∈ 1, . . . , d and any t ∈ R, X i(t) =

R f (t − s)Li(ds), ‘‘⇐’’ is obvious.‘‘⇒’’: Expression (5) also has an alternative form νf (B) =

R\f−1(0) ρ

(1/f (s))B

ds for any B ∈ B(Rd). Independence of

X1(t) · · · , Xd(t) implies that the Lévymeasure ofX(t) is supported by the union of the coordinate axes (see E 12.10 onp. 67 ofSato (1999)). Therefore, ν I

f (B) =

R\f−1(0) ρ I(1/f (s))B

ds = 0 for all I ⊂ 1, . . . , dwith |I| ≥ 2 and all B ∈ B((R \ 0)|I|).

The nonnegativity of the integrand implies that ρ I(1/f (s))B

= 0 holds for all s ∈ R\ f −1(0), all I ⊂ 1, . . . , dwith |I| ≥ 2

and all B ∈ B(R \ 0)|I|

, which is equivalent to ρ I

B= 0 for all I ⊂ 1, . . . , d with |I| ≥ 2 and all B ∈ B

(R \ 0)|I|

.

That is, the Lévy measure ρ is supported by the union of the coordinate axes. A result of Sato (1999) (E 12.10 on p.67) nowallows us to conclude that L1, . . . , Ld are independent.

Remark 3. The independent Lévy copula in the multivariate case is given by the form (see Cont and Tankov (2004) andKallsen and Tankov (2006))

C⊥(x1, . . . , xd) =d

i=1

xij=i

1∞(xj) for any (x1, . . . , xd) ∈ Rd.

To analyze the MA driven by a Lévy process with a completely jump dependent (or comonotonic) Lévy measure, we firstrecall the notion of an ordered set in Kallsen and Tankov (2006). Recall that a subset O of Rd is called ordered if, for any twovectors u, v ∈ O, either uk ≤ vk, k = 1, . . . , d or uk ≥ vk, k = 1, . . . , d. Similarly, O is called strictly ordered if, for any twodifferent vectors u, v ∈ O, either uk < vk, k = 1, . . . , d or uk > vk, k = 1, . . . , d.

For convenience, we say a Lévy measure ν on Rd is completely dependent or comonotonic, if there exists a strictly orderedsubset O such that ν(Rd

\ O) = 0.Let ν be a Lévy measure on Rd. By Theorem 4.4 of Kallsen and Tankov (2006), the Lévy measure ν is supported by an

ordered set O (O ⊂ Rd) if and only if there exists a Lévy copula of ν given by

C||(x1, . . . , xd) = min|x1|, . . . , |xd|1K (x1, . . . , xd)d

i=1

sgn(xi) (19)

with K = x ∈ Rd: sgn(x1) = · · · = sgn(xd), for any (x1, . . . , xd) ∈ Rd.

Theorem 2 shows that if the Lévy measure of L is completely dependent, then the Lévy measure of the law of X(t) iscompletely dependent.

Theorem 2. Let X be an MA driven by a d-dimensional Lévy process L whose Lévy measure is symmetric. Suppose all theone-dimensional marginal Lévy measures of the Lévy measure of L are equal. Under Condition 1, if the Lévy copula of L is givenby (19), then there exists a Lévy copula of X(t) given by (19). If, in addition, the Lévy measure of Li is infinite and has no atomsfor any i ∈ 1, . . . , d (or alternatively, R \ f −1(0) is an unbounded set and the Lévy measure of Li has no atoms for anyi ∈ 1, . . . , d), (19) is the unique Lévy copula of X(t).

Proof. By standard calculations from equality (10), we obtain that CI

(Ufi(xi))i∈I= C||

(Ufi(xi))i∈I

for any non-empty

I ⊂ 1, . . . , d and any x ∈ Rd. Since C|| defines a Lévy copula (see the proof of Theorem 4.4 of Kallsen and Tankov (2006)),this proves that there exists a Lévy copula of the law of X(t) given by (19). If the Lévy measure of Li is infinite and has noatoms for any i ∈ 1, . . . , d (or alternatively,R\ f −1(0) is an unbounded set and the Lévymeasure of Li has no atoms for anyi ∈ 1, . . . , d), by (7), we have the tail integral Ufi is continuous and satisfies limx→0 Ufi(x) = ∞. According to Theorem 4.4of Kallsen and Tankov (2006), C|| is the unique Lévy copula of X(t).

S. Zhang / Journal of the Korean Statistical Society 41 (2012) 555–562 561

A Lévy copula C is called homogeneous (of order 1), if

C(ru1, . . . , rud) = rC(u1, . . . , ud) (20)

for any r > 0 and any (u1, . . . , ud) ∈ Rd.By Theorem 4 of Barndorff-Nielsen and Lindner (2007), if the Lévymeasure of L is finite, concentrated on (0,∞)d, and has

a homogeneous Lévy copula, then the Lévy copula of Lmust have expression (19). Thereforewe have the following corollary.

Corollary 1. Let X be anMA driven by a d-dimensional subordinator L, whose Lévymeasure is finite and has a homogeneous Lévycopula. Suppose all the one-dimensional marginal Lévy measures of the Lévy measure of L are equal. Then, under Condition 1,there exists a Lévy copula of X(t) given by (19). If, in addition, R+ \ f −1(0) is an unbounded set and the Lévy measure of Li hasno atoms for any i ∈ 1, . . . , d, (19) is the unique Lévy copula of X(t).

For the class of multivariate α-stable distributions (see Samorodnitsky and Taqqu (1994)), the Lévy measure ν satisfies

ν(B) =Sλ(dξ)

∞

01B(rξ)

drr1+α

, B ∈ B(Rd), (21)

where λ is a finite measure on the unit sphere S. We shall give some stable invariance and dependence invariance under thetransform of the MA.

Theorem 3. Let X be an MA driven by a d-dimensional α-stable Lévy process L and let α ∈ (0, 2). Under Condition 1, the lawof X(t) is α-stable. Moreover, if d ≥ 2, then the Lévy copula of the Lévy measure of the law of X(t) is same to that of the Lévymeasure of L, i.e., C = C on Rd, where the common Lévy copula is homogeneous.

Proof. If L is a d-dimensional α-stable Lévy process, then its Lévy measure ρ is given by (21). Then, by (14.7) of Sato (1999),(5) can be re-expressed as

νf (B) =

R

f (s)

α Sλ(dξ)

∞

01B(rξ)

drr1+α

, B ∈ B(Rd), (22)

which is obviously the Lévy measure of an α-stable law.If L is a d-dimensional (d ≥ 2) α-stable Lévy process, then all its components Li (i = 1, . . . , d) are one-dimensional

α-stable Lévy processes (see Theorem 4.6 of Kallsen and Tankov (2006)). For each i ∈ 1, . . . , d, we assume the tail integralof Li has form (13). Then, for any i ∈ 1, . . . , d, the tail integral of the law of X i(t) is given by (14). If C be the Lévy copulaof the d-dimensional α-stable Lévy process L, then by Theorem 4.6 of Kallsen and Tankov (2006), C satisfies property (20).From (15), we obtain that for any non-empty I ⊂ 1, . . . , d, the Lévy copula of the law of X(t) satisfies

CI

(ui)i∈I=

R( f (s))α ds

−1C I(ui)i∈I

R( f (s))α ds = C I(ui)i∈I

for u = (u1, . . . , ud) ∈ Rd. That is, C = C on Rd. This completes the whole proof.

All the above results in this section are applicable for the O–U process (Example 2). Furthermore, for the O–U process,we can obtain the result more elaborate than Theorem 3 as follows.

Theorem 4. Let X be an O–U process driven by a d-dimensional Lévy process L and let α ∈ (0, 2). Suppose the Lévy measure ofL satisfies (17). Then, the following statements are equivalent.

(i) L is an α-stable Lévy process.(ii) The transition law of X is α-stable.(iii) The stationary law of X is α-stable.

Moreover, if d ≥ 2, then all three Lévy copulas (which are of the Lévymeasure of L, of the transition law of X and of the stationarylaw of X) are same on Rd, where the common Lévy copula is homogeneous.

Proof. It follows from Theorem 3 that ‘‘(i)⇒ (ii)’’, ‘‘(i)⇒ (iii)’’ and that all three Lévy copulas are same hold obviously. Notethat the kernel functions of the stationary law and the transition law are given respectively by (16) and (18). Then, since thestationary law is the limit of the transition law as ∆→∞, by (22), we obtain that ‘‘(ii)⇒ (iii)’’ hold. If the Lévy measure ofthe stationary law is given by (21), then by Theorems 14 and 41 of Rocha-Arteaga and Sato (2003), the Lévy measure of theL is given by

ρ(B) = µα

Sλ(dξ)

∞

01B(rξ)

drr1+α

, B ∈ B(Rd),

which is a Lévy measure of an α-stable Lévy process. This proves the ‘‘(iii)⇒ (i)’’ part. We complete the whole proof of thetheorem.

562 S. Zhang / Journal of the Korean Statistical Society 41 (2012) 555–562

Acknowledgments

I am grateful to the anonymous reviewers for their valuable comments which resulted in an improved version of thearticle. This research is supported by the National Natural Science Foundation of China (grant number 10901100) and theScience & Technology Program of Shanghai Maritime University (grant number 20100135).

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