The distribution of the Rosenblatt process { Examples of...

The distribution of the Rosenblatt process– Examples of distributions in the Thorin class –

Makoto Maejima

Keio University, Yokohama, Japan

(Joint work with Ciprian A. Tudor at Lille)

July 18, 20137th International Conference on Levy processes, Wroclaw

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Outline

1 The Rosenblatt process

2 The Thorin class

3 Results

4 The non-symmetric Rosenblatt process

5 The unimodality of the Rosenblatt distribution

6 Stochastic integral representations with respect to Levy process of theRosenblatt distribution

7 Other recent examples of GGC distributions

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The Rosenblatt process

Let 0 < D < 12 . The Rosenblatt process is defined, for t ≥ 0, as

ZD(t) =C(D)∫ ′

R2

(∫ t

0(u− s1)

−(1+D)/2+ (u− s2)

−(1+D)/2+ du

)dB(s1)dB(s2),

where {B(s), s ∈ R} is a standard Brownian motion,∫ ′

R2 is the integralover R2 except the hyperplane s1 = s2 and C(D) is a normalizingconstant. The distribution of ZD(1) is called the Rosenblatt distribution.

The Rosenblatt process is H-selfsimilar with H = 1−D and hasstationary increments.The Rosenblatt process lives in the so-called second Wiener chaos.Consequently, it is not a Gaussian process.In the last few years, this stochastic process has been the object of severalpapers. (See Pipiras-Taqqu(2010), Tudor(2008), Tudor-Viens(2009),Veillette-Taqqu(2012) among others.)

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The Rosenblatt distribution is the first example of non-Gaussian limitingdistribution of normalized partial sums of some strongly dependentstationary random variables discovered by Rosenblatt(1961), but very fewthings had been known about its distributional properties.

The Rosenblatt distribution is known to be infinitely divisible.Recently, in the work by Veillette-Taqqu(2012), they computed, amongothers, its Levy measure, its cumulants and moments. They also derivednumerically the shape of the density.

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The Thorin class (I)

We first define the Thorin class of probability distributions on R+.Originally this class was studied by Thorin(1977, two papers) when hewanted to prove the infinite divisibility of the Pareto distribution and ofthe lognormal distribution. The Thorin class on R+, denoted by T (R+), isthe smallest class of distributions on R+ that contains all gammadistributions and is closed under convolution and weak convergence.A probability distribution in T (R+) is called generalized gammaconvolution (GGC).

This class was extended to Rd by Barndorff-Nielsen-M.-Sato(2006) asfollows: Call Γx an elementary gamma random variable in Rd if x is anon-random non-zero vector in Rd and Γ is a gamma random variable onR+. Then the Thorin class on Rd, denoted by T (Rd), is defined as thesmallest class of distributions on Rd that contains all elementary gammadistributions on Rd and is closed under convolution and weak convergence.(The Thorin class on R is already defined in Thorin(1978) as the name ofthe extended generalized gamma convolutions (EGGC).)

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This class is a subclass of the class of selfdecomposable distributions, butit is surprisingly rich.

1 L. Bondesson, Generalized Gamma Convolutions and Related Classesof Distributions and Densities. Lecture Notes in Statistics 76,Springer, 1992.

2 L.F. James, B. Roynette and M. Yor, Generalized gammaconvolutions, Dirichlet means, Thorin measures, with explicitexamples, Probability Surveys 5 (2008), 341-415.

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The Thorin class (II)

For any infinitely divisible distribution µ on Rd, we have the followingLevy-Khintchine representation of the characteristic function µ(z), z ∈ Rd,of µ:

µ(z) = exp{−1

2〈z, Az〉+ i〈γ, z〉+∫

Rd

(ei〈z,x〉 − 1− i〈z, x〉

1 + |x|2

)ν(dx)

},

where A is a symmetric nonnegative-definite d× d matrix, ν is a measureon Rd (called Levy measure) satisfying

ν({0}) = 0 and

∫Rd

(1 ∧ |x|2)ν(dx) < ∞,

and γ ∈ Rd.

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• Polar decomposition of Levy measure ν

ν(B) =∫

Sλ(dξ)

∫ ∞

01B(rξ)νξ(dr), B ∈ B(Rd \ {0}),

where λ is a measure on S = {ξ ∈ Rd : |ξ| = 1},{νξ : ξ ∈ S} is a family of positive measures on (0,∞),We call νξ the radial component of ν.

When νξ is differentiable, its density `ξ(r) (νξ(dr) = `ξ(r)dr) is called theLevy density.

µ ∈ T (Rd) if and only if r`ξ(r) is completely monotone in r ∈ (0,∞).

Namely,

`ξ(r) =kξ(r)

r

and kξ(r) is completely monotone.

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Representation of ZD(t)Let W be a complex-valued Gaussian random measure on R such that forBorel sets in R, A, B,Aj , E[W (A)] = 0, E[W (A)W (B)] = the Lebesgue

measure of A ∩B, W(⋃n

j=1 Aj

)=∑n

j=1 W (Aj) for mutually disjoint

sets A1, ..., An and W (A) = W (−A).Let

HD ={

h : h is a complex-valued function on R, h(x) = h(−x),∫R

h(x)2|x|D−1dx < ∞}

and for every t ≥ 0 define an integral operator At by

Ath(x) = C(D)∫ ∞

−∞

eit(x−y) − 1i(x− y)

h(y)|y|D−1dy, h ∈ HD.

Since At is a self-adjoint Hilbert-Schmidt operator (seeDobrushin-Major(1979)), all eigenvalues λn(t), n = 1, 2, ..., are real andsatisfy

∑∞n=1 λ2

n(t) < ∞.Maejima (Keio) Thorin class July 18, 2013 9 / 29

Our first theorem is as follows.

Theorem (1)

For every t1, ..., td ≥ 0,

(ZD(t1), ..., ZD(td))d=

( ∞∑n=1

λn(t1)(ε2n − 1), ...,

∞∑n=1

λn(td)(ε2n − 1)

),

where {εn} are i.i.d. N(0, 1) random variables.

The case d = 1 was shown by Taqqu (see Proposition 2 ofDobrushin-Major(1979)).

The proof is enough to extend the idea of Taqqu from one dimension tomulti-dimension.

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Theorem (2)

For every t1, ..., td ≥ 0, the law of (ZD(t1), ..., ZD(td)) belongs to T (Rd).

Proof. By Theorem (1),

(ZD(t1),..., ZD(td))

d=

( ∞∑n=1

λn(t1)(ε2n − 1), ...,

∞∑n=1

λn(td)(ε2n − 1)

)

=∞∑

n=1

ε2n(λn(t1), ..., λn(td))−

( ∞∑n=1

λn(t1), ...,∞∑

n=1

λn(td)

),

where ε2n(λn(t1), ..., λn(td)), n = 1, 2, ..., are the elementary gamma

random variables in Rd. Since they are independent, by the properties ofthe class T (Rd) that the class is closed under convolution and weakconvergence, we see that

∑∞n=1 ε2

n(λn(t1), ..., λn(td)) belongs to T (Rd),and so does (ZD(t1), ..., ZD(td)). This completes the proof.

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The non-symmetric Rosenblatt process

In M.-Tudor(2012), we extended the Rosenblatt process and introducedthe non-symmetric Rosenblatt process as follows.Let 0 < D1 6= D2 < 1

2 . The non-symmetric Rosenblatt process is definedas

ZD1,D2(t) = C(D1, D2)∫ ′

R2

(∫ t

0(u− s1)

−(1+D1)/2+ (u− ss)

−(1+D2)/2+ du

)dB(s1)dB(s2).

This process is selfsimilar with stationary increments in the second Wienerchaos. When D1 = D2 = D, it is the Rosenblatt process ZD(t).

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Theorem

For every t1, ..., td ≥ 0, the law of (ZD1,D2(t1), ..., ZD1,D2(td)) belongs toT (Rd).

Proof. For α1, ...αd ∈ R,

α1ZD1,D2(t1) + · · ·+ αdZD1,D2(td)

d=∫ ′′

R2

(α1

eit1(x1+x2) − 1i(x1 + x2)

+ · · ·+ αdeitd(x1+x2) − 1

i(x1 + x2)

)|x1|(D−1)/2|x2|(D−1)/2W (dx1)W (dx2)

d=∞∑

n=1

λ(D1,D2)n (t1, ..., td)(ε2

n − 1),

where∫ ′′

R2 is the integral over R2 except the hyperplanes x1 = ±x2 and

λ(D1,D2)n (t1, ..., td) are eigenvalues of some integral operator. This can be

shown by extending the idea for d = 1 of Proposition 2 ofDobrushin-Major(1979) to the case d ≥ 1.

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The unimodality of the Rosenblatt distribution

Let us consider the Rosenblatt distribution when d = 1.(The following are the same for the non-symmetric Rosenblattdistribution.)

We have seen that their distributions belong to the Thorin class T (R).

Thus, the Rosenblatt distribution is selfdecomposable, and hence it hasthe density and unimodal, which was suggested numerically byVeillette-Taqqu(2012).

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Stochastic integral representations with respect to Levyprocess of the Rosenblatt distribution

The Rosenblatt distribution is represented by double Wiener-Ito integral.However, the distributions in T (R) have several stochastic integralrepresentations with respect to Levy processes.We regard them as members of the class of selfdecomposabledistributions, which is a larger class than the Thorin class.This allows us to obtain a new result related to the Rosenblatt distribution.

We know that any selfdecomposable random variable X has the stochasticintegral representation with respect to some Levy process {Zt} in law.

Namely, Xd=∫∞0 e−tdZt. However, for the Rosenblatt distribution, we can

give an explicit form of {Zt}.

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Aoyama-M.-Ueda(2011) showed that if {γt,λ, t ≥ 0} is a gamma processwith parameter λ > 0, {N(t), t ≥ 0} is a Poisson process with unit rateand they are independent, then

γ 12, 12

d=∫ ∞

0e−tdγN( 1

2t), 1

2.

LetYt = γN( 1

2t), 1

2− t.

Note that {Yt, t ≥ 0} is a Levy process. Then we have

ε2n − 1 d= γ 1

2, 12− 1 d=

∫ ∞

0e−tdYt,

Let {Y (n)t } be independent copies of {Yt}. Then

ZDd=

∞∑n=1

λn(ε2n − 1) d=

∫ ∞

0e−td

( ∞∑n=1

λnY(n)t

)=:∫ ∞

0e−tdZt.

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Remark∑∞n=1 λnY

(n)t is convergent a.s. and in L2 because

∞∑n=1

E

[(λnY

(n)t

)2]

= E[Y 2

t

] ∞∑n=1

λ2n < ∞.

Remark

Since {Y (n)t }, n = 1, 2, ..., are independent and identically distributed Levy

processes, their infinite weighted sum {Zt} is a Levy process.

We thus finally have the following theorem.

Theorem

ZDd=∫ ∞

0e−tdZt,

where {Zt} is a Levy process defined above.

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Other recent examples of GGC.

Bertoin-Fujita-Roynette-Yor(2006)(Random excursion of Bessel processes)Let {Rt, t ≥ 0} be a Bessel process with R0 = 0, with dimensiond = 2(1− α). (0 < α < 1, equivalently 0 < d < 2.)When α = 1

2 , {Rt} is a Brownian motion. Let

g(α)t := sup{s ≤ t : Rs = 0},

d(α)t := inf{s ≥ t : Rs = 0}

and∆(α)

t := d(α)t − g

(α)t ,

which is the length of the excursion above 0, straddling t, for the process{Ru, u ≥ 0}, and let ε be a standard exponential random variable

independent of {Ru, u ≥ 0}. Let ∆α := ∆(α)ε . Then

L(∆α) ∈ T (R+)(⊂ L(R+)).

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In Bertoin-Fujita-Roynette-Yor(2006), only “∈ L(R)” is mentioned.However, they actually showed that

E[e−λ∆α

]= exp

{−(1− α)

∫ ∞

0

(1− e−λx

) E[e−xGα ]x

dx

}, λ > 0,

with a random variable Gα. (The density function of Gα is explicitlygiven.) Since k(x) := E[e−xGα ] is completely monotone by Bernstein’stheorem, L(∆α) belongs not only to L(R+) but also to T (R+).

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Handa(2012)(Continuous state branching processes with immigration)

CBCI-process with quadruplet (a, b, n, δ) :Continuous state Branching process with Continuous Immigrationwith the generator

Lδf(x) = axf ′′(x)−bxf ′(x)+x

∫ ∞

0[f(x+y)−f(x)−yf ′(x)]n(dy)+δf ′(x),

where n is a measure on (0,∞) satisfying∫∞0 (y ∧ y2)n(dy) < ∞.

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If we restrict ourselves to GGC on R+, µ ∈ T (R+) has the Laplacetransform:

π(s) :=∫ ∞

0e−sxµ(dx), s > 0,

= exp{−γs−

∫ ∞

0(1− e−sx)ν(dx)

},

where γ ≥ 0 and∫∞0 (1 ∧ x)ν(dx) < ∞.

Note that

µ ∈ T (R+) ⇔ ν(dx) = `(x)dx and x`(x) is completely monotone on (0,∞).

Since the completely monotone function is the Laplace transform of someσ-finite and positive measure (let’s say, σ) by Bernstein’s theorem, we have

x`(x) =∫ ∞

0e−xyσ(dy).

When ` is the Levy density of GGC, we call σ the Thorin measure.Maejima (Keio) Thorin class July 18, 2013 21 / 29

Recall

π(s) :=∫ ∞

0e−sxµ(dx), s > 0,

= exp{−γs−

∫ ∞

0(1− e−sx)ν(dx)

}= exp

{−γs−

∫ ∞

0(1− e−sx)

1x

(∫ ∞

0e−xyσ(dy)

)dx

}.

Since GGC on R+ is determined by γ and σ, we call it the GGC with pair(γ, σ).

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Theorem

Let γ ≥ 0 and suppose that σ is a non-zero Thorn measure.

(1) There exist (a, b,M) such that

γ +∫

1s + u

σ(du) =1

as + b +∫

ss+uM(du)

, s > 0.

(2) Any GGC with pair (γ, σ) is a unique stationary solution ofCBCI-process with quadruplet (a, b, n, 1), where n is a measure on (0,∞)defined by

n(dy) =(∫ ∞

0u2e−yuM(du)

)dy.

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Takemura-Tomisaki(2012)(Levy density of inverse local time of some diffusionprocesses)

Example (Also, Shilling-Song-Vondracek(2010),p.201)

Let I = (0,∞) and −1 < ν < 0.

Let G(ν) = 12

d2

dx2 + 2ν+12x

ddx .

Assume 0 is reflecting.D(ν) : the diffusion process on I with the generator G(ν)

n(ν) : the Levy density of the inverse local time at 0 for D(ν)

=⇒ n(ν)(x) = C 1xx−|ν| (GGC)

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Example

Let I = (0,∞) and −1 < ν < 0.

G(ν) = 2x d2

dx2 + (2ν + 2) ddx

D(ν) : the diffusion process with the generator G(ν)

and the end point 0 being reflecting.n(ν) : the Levy density of the inverse local time at 0 for D(ν)

=⇒ n(ν)(x) = C 1xx−|ν| (GGC)

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Example

Let −1 < ν < 1 and β > 0. Let

G(ν,β) =12

d2

dx2+{

12x

+√

2βK ′

ν(√

2βx)Kν(

√2βx)

}d

dx,

where Kν(x) is the modified Bessel function.D(ν,β) : the diffusion process on I with the generator G(ν,β)

and the end point 0 being reflecting.n(ν,β) : the Levy density of the inverse local time at 0 for D(ν,β)

=⇒ n(ν,β)(x) = C 1xx−|ν|e−βx. (GGC)

(When ν = 0, Shilling-Song-Vondracek(2010),p.202.)

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Example (Shilling-Song-Vondracek(2010),p.201)

Let 0 < ν < 1 and β > 0. Let

G(ν,β) =12

d2

dx2+{

β − 12x

+√

2βK ′

ν(√

2βx)Kν(

√2βx)

}d

dx,

where Kν(x) is the modified Bessel function.D(ν,β) : the diffusion process on I with the generator G(ν,β)

and the end point 0 being reflecting.n(ν,β) : the Levy density of the inverse local time at 0 for D(ν,β)

=⇒ n(ν,β)(x) = C 1xx−νe−βx. (GGC)

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Example

Let −1 < ν < 1 and β > 0. Let

G(ν,β) = 2xd2

dx2+ 2

{1 +

√2βx

K ′ν(√

2βx)Kν(

√2βx)

}d

dx.

D(ν,β) : the diffusion process with the generator G(ν,β)

and the end point 0 being reflecting.n(ν,β) : the Levy density of the inverse local time at 0 for D(ν,β)

=⇒ n(ν,β)(x) = C 1xx−|ν|e−βx. (GGC)

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Thank you very much!

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