FREE GENERALIZED GAMMA CONVOLUTIONS. Victor Pérez-Abreu and Noriyoshi Sakuma. Comunicación del CIMAT No I-08-19/ (PE /CIMAT)

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1 FREE GENERALIZED GAMMA CONVOLUTIONS Victor Pérez-Abreu and Noriyoshi Sakuma Comunicación del CIMAT No I-8-9/ PE /CIMAT

2 FREE GENERALIZED GAMMA CONVOLUTIONS VICTOR PÉREZ-ABREU AND NORIYOSHI SAKUMA 2 Abstract. The so-called Bercovici-Pata bijection maps the set of classical infinitely divisible laws to the set of free infinitely divisible laws. The purpose of this work is to study the free infinitely divisible laws corresponding to the classical Generalized Gamma Convolutions GGC. Characterizations of their free cumulant transforms are derived as well as free integral representations with respect to the free Gamma process. A random matrix model for free GGC is built consisting of matrix random integrals with respect to a classical matrix Gamma process. Nested subclasses of free GGC are shown to converge to the free stable class of distributions.. Introduction Generalized Gamma Convolutions GGC is the smallest class T R + of classical infinitely divisible distributions on R + that contains all Gamma distributions and that is closed under classical convolution and weak convergence. This class was introduced by Thorin [6], [7] and further studied by Bondesson [8]. Thorin [8] also considered the smallest class of distributions on the real line which contains all distributions in T R + and is closed under convolution, convergence and reflection. We denote this class by T R and called it the Thorin class of distributions on R. Let PR be set of probability measures on R and I R the class of all classical infinitely divisible distributions in PR. If µ PR, ˆµz denotes its Fourier transform and when µ I R we denote by Cµz its classical cumulant function or Lévy exponent i.e. Cµz = log ˆµz. A probability measure µ PR is in I R if and only if its classical cumulant function has the Lévy-Khintchine representation : Cµz = 2 az2 + iηz + e izx izx { x } νdx, z R, R where a, γ R and ν the so called Lévy measure is a measure satisfying ν{} = and R x 2 <. The triplet a, η, ν is uniquely determined and is called - characteristic triplet or simply -triplet. When x νdx <, we speak of the drift R type Lévy Khintchine representation 2 Cµz = 2 az2 + iη z + e izx νdxz R, Date: September 5, 28. Part of this work was done while the author was visiting Keio University during the Fall of 27. He acknowledges the support and hospitality of the mathematics department of this university. 2 Part of this work was done while the author was visiting CIMAT during the Spring of 28. He sincerely appreciates the support and hospitality of CIMAT. He is supported by Japan Society for the Promotion of Science. R

3 where η is the drift of µ and is given by η = η xνdx. We write { x } I log { R = µ I R; log x µdx < } and refer to Sato [3] for basic facts about classical infinitely divisible distributions. R Bondesson [8] proved that a positive random variable Y with classical law L Y = µ -without translation term- belongs to T R + if and only if there exists a positive Radon measure U µ on, such that 3 C µz = = e izx dx e xs U µ ds x log + iz U µ ds s with log x U µdx < and U µ dx <. The measure U x µ is called the Thorin measure of µ. So, the -triplet of µ is,, ν µ where the Lévy measure is concentrated on, and such that 4 ν µ dx = dx x e xs U µ ds. It is known that the class T R + is characterized by Wiener-Gamma representations, i.e., random integral representations with respect to the standard one-dimensional Gamma process see [7], []. Specifically, a positive random variable Y belongs to T R + if and only if there is a Borel function h : R + R + with 5 log + htdt <, such that Y d = Y h has the Wiener-Gamma integral representation 6 Y h L = hudγ u, where γ t ; t is the standard Gamma process with Lévy measure νdx = e x dx. x Moreover, C Y z = log + iz U h µ h dx, x where Uµ h denotes the image of Lebesgue measure on, under the application : s /hs. That is, 7 e x hs ds = e xz Uµ h dz, x >. The function h is called the Thorin function of Y and is obtained as follows. Let F Uµ x = x U µdy for x and let F U µ s be the right continuous inverse of F U µ s in the sense of composition of functions, that is F U µ s = inf{t > ; F Uµ t s} for s. Then, hs = /F U µ s for s. Moreover we have the following alternative 2

4 expression for the cumulant function of µ 8 C µz = ds e izx e x/hs x In the above equation, if {t > ; F Uµ t s} = ϕ, x/hs =. Remark.. If Y is a GGC random variable, we write Y h respectively µ h to indicate that it has the integral representation 6 and write µ h = L Y h. We have excluded from the above discussion the case of non-zero drift, which is easily incorporated by considering nonzero drift c in the -triplet c,, ν µ. Many well known distributions belong to T R +. The positive α-stable distributions, < α <, are GGC with hs = {sθγα + } α for a θ >. In particular, for the /2 stable distribution, hs = 4 s 2 π. First passage time distribution, Beta distribution of the second kind, lognormal and Pareto are also GGC, see []. As for distributions in T R, there is a another random integral representation approach recently presented in Barndorff-Nielsen et. al [], who also considered the multivariate case. We recall that if X t ; t is a -Lévy process and f : [a, b] R is a continuous function defined on an interval [a, b] in [,, then the stochastic integral ftdx [a,b] t may be defined as the limit in probability of approximating Riemann sums. Moreover, if f is continuous function defined on [,, ftdx [a, t may be as the limit in probability of ftdx [a,b t when b. For stochastic integrals of nonrandom functions with respect to general additive processes we refer to Sato [4]. It is shown in [], that for any µ in I R, the mapping Υ given by 9 Υ µ = L log t dxµ t, is always defined, where X µ t is a Lévy process with L X µ = µ. Moreover T R = Υ L R, where L R is the class of -selfdecomposable distributions in R : µ L R if for any b, there exists ρ b PR such that ˆµz = ˆµbz ˆρ b z. Furthermore, it is shown in [] that a random variable Y belongs to T R if and only if there exists µ Ilog R such that Y = e tdx µ t where the function e t is the inverse of the incomplete gamma function e x = x e s s ds and X µ t is a Lévy process with L X µ = µ. In the study of relations between classical and free infinite divisibility, Bercovici and Pata [5] introduced a bijection Λ between the set I R of classical infinitely divisible laws and the set I R of free infinitely divisible laws. A new approach to this bijection was recently proposed by Benaych-Georges [4] and Cabanal-Duvillard [9]. They construct random matrix ensembles associated to classical one-dimensional infinitely divisible laws whose empirical spectral laws converge to their corresponding free infinitely divisible laws under Λ. Recall that an ensemble of random matrices is a sequence M d d where for each d, M d is a d d matrix with random entries. The random spectral dx. 3

5 measure or empirical spectral law µ M d d of M d is defined as the uniform distribution on the spectrum λ M d,..., λ M d d, that is, µ M d d = d d. An ensemble M d d is a i= δ λ Md i Random Matrix Model RMM for a probability measure µ, if µ M d d converges to µ weakly in probability as d. It is shown in [4], [9] that for any µ I R, there exists a random matrix model M d d for Λµ, which is constructed from µ. These papers generalize the pioneering work by Wigner who connects Gaussian and semicircle laws throughout the Gaussian Unitary Ensemble of random matrices. The purpose of this work is to study the free infinitely divisible laws FGGC corresponding to the image of Λ of classical Generalized Gamma Convolutions and their corresponding random matrix models. We start in Section 2 by recalling facts and notation about the free cumulant function, the Bercovici-Pata bijection, free Lévy process and their random integrals. In Section 3 we prove a characterization of the free cumulant transform of a FGGC analogous to the classical cumulant transform 3. Furthermore, we derive free integral representations with respect to the free Gamma process and a Lévy process similar to 6 and, respectively. In Section 4 we construct random matrix models for FGGC. They are given as classical matrix random integrals of Wiener-Gamma type similar to 6, with respect to an appropriate classical matrix Gamma process. Finally, in Section 5 we point out some facts on nested subclasses of ΛT R and their limits, analogous to the recent results for the classical convolution case study in Maejima and Sato []. 2. Preliminaries on Free infinite divisibility The Cauchy-Stieltjes transform of a probability measure µ on R is defined by G µ z = z t µdt, z C+. R The function F µ z = /G µ z has right inverse F µ z on the region Γ η,m for some M > and η >, where Γ η,m := {z C : Re z < ηim z, Im z > M}; see Bercovici and Voiculescu [6]. Following Barndorff-Nielsen and Thorbjørnsen [3], the free cumulant transform C µ of µ is defined by C µ z = zf µ z z Γ η,m. A probability measure µ on R is -free infinitely divisible if and only if C µ z has an analytic continuation to C. We denote by I R the class of all free infinitely divisible distributions. In complete analogy to the classical case, the free Lévy-Khintchine characterization establishes that a probability measure µ belongs to I R if only if C µ z = η µ z + a µ z 2 + tz tz [,] t ν µ dt, z C, R where a µ R +, η µ R and the Lévy measure ν µ is a measure satisfying ν µ {} = and R x 2 ν µ dx <. In this case, the -triplet a µ, ν µ, η µ is uniquely determined by µ and is called the -characteristic triplet or -triplet for µ, see [3] [6]. 4

6 Bercovici and Pata [5] introduced a bijection Λ between classical and free infinitely divisible distributions. It is such that if µ I R has -characteristic triplet a µ, ν µ, η µ, then Λµ is the free infinitely divisible distribution with -triplet a µ, ν µ, η µ. Remark 2.. If µ I R and its Lévy measure ν µ satisfies x ν R µdx <, then for z C C µ z = η µ z + a µ z 2 + R tz tz [,] t ν µ dt = η µ xνdx z + a µ z 2 + { x } R tz ν µ dt 2 = η µz + a µ z 2 + tz ν µ dt, R where η µ = η µ { x } xν µdx. We call this representation the drift type -cumulant of µ I R and η µ is the -drift. By Bercovici-Pata bijection, if µ I R has -drift type triplet a µ, ν µ, η µ then the -drift type triplet of Λµ is also a µ, ν µ, η µ. We summarize some properties of the Bercovici-Pata bijection in the following result see [3], [5], [6]. Proposition 2.2. The map Λ : I R I R has the following properties. Λµ ρ = Λµ Λρ for any µ, ρ PR. 2 Let δ a be Dirac measure at a. Λδ a = δ a for a R. So Λ is preserved under affine transforms, i.e. ΛD c µ δ a = D c Λµ δ a for any b > and a R where D c µ means the spectral distribution of the operator cx with µ = LX. 3 Λ is a homeomorphism w.r.t. weak convergence i.e. µ n µ if and only if Λµ n Λµ in weak convergence. For a classical random variable X or a stochastic process X t, we write ΛX and ΛX t as a short notation for ΛL X and ΛL X t. Barndorff-Nielsen and Thorbjørnsen [3] introduced free selfdecomposable distribution. A probability measure µ on R is free selfdecomposable -selfdecomposable if, for any b,, there exists ρ b PR such that µ = D b µ ρ b. We denote by L R the class of all free self-decomposable distributions on R. We refer to Sakuma [5] for a detailed study of -selfdecomposable distributions. As in the classical case, free Lévy process and their free integrals can be considered with respect to the convolution. Given a free random variable Z, we denote by L Z its spectral distribution. Following [3], we say that a process Z t ; t of selfadjoint operators affiliated with a W -probability space A, τ, is a free Lévy process in law if it satisfies the following four conditions: Z = 2 Whenever n N and t < t < t n, the increments Z t, Z t Z t, Z t2 Z t,, Z tn Z tn, 5

7 are freely independent random variables. 3 For any s, t in [,, L Z s+t Z s does not depend on s. 4 For any s [,, L Z s+t Z s converge weakly to δ, as t. For any compact interval [a, b] [, and any continuous function f : [a, b] R, the random integral b ftdz a t exists as the limit in probability of approximating Riemann sums. The following result summarizes the connection between classical and free random integrals, see [3]. Proposition 2.3. Let X t be a classical Lévy process and Z t be a free Lévy process with marginal distribution µ t and Λµ t, respectively. Then for any [a, b] [, and any continuous function f : [a, b] R, the laws L b ftdx a t and L b ftdz a t are -infinitely divisible and -infinitely divisible, respectively. Moreover, b b 3 L ftdz t = Λ L ftdx t. a In particular, if Y is a free selfdecomposable random variable, there exists a free Lévy process Z t such that LZ = µ, R\, log+ t ν µdt < and L Y = e t dz t. 3. Free Generalized Gamma Convolutions When γ is the classical gamma distribution, we call Λγ the free gamma distribution. If γ t ; t is the standard Gamma process, the free Lévy process Λγ t ; t is called the free standard Gamma process. We say that a probability distribution λ is Free Generalized Gamma Convolution FGGC resp. Free Thorin if there is a classical GGC resp. Thorin µ such that λ = Λµ. We denote by T R + = ΛT R + and T R = ΛT R the classes of FGGC and Free Thorin class respectively. It follows trivially from Proposition 2.2, that T R + is the smallest class that contains all free Gamma distributions and that is closed under -convolution and convergence, while T R is the smallest class on the real line R which contains T R + and is closed under convolution, convergence and reflection. The following result is a characterization of the free cumulant transform of distributions in T R + in terms of the Cauchy transform of the exponential distribution. Theorem 3.. A probability measure λ in R + is FGGC without drift term if and only if there exists a Borel function h : R + R + satisfying 5 such that λ has free cumulant transform 4 C λ z = hsg E hs z ds z C, where G Ea is the Cauchy transform of the exponential law with mean /a, i.e. 5 G Ea z = a ae ax z x dx z C+. 6

8 Alternatively, a probability measure λ in R + is FGGC without drift term if and only if there is a Thorin measure U µ h such that 6 C λ z = s G Esz U µ ds z C. Proof. For any t, the Lévy measure of γ t has finite first moment. We work with the drift type representation 2 with η µ = a µ =. First, since γ t and Λγ t have the same characteristic and -triplet, from 2, the free cumulant transform of Λγ t is obtained as e x C Λγt z = t xz x dx 7 = t e x z x dx = tg E z z C. Next, by Remark., a probability measure λ without drift term belongs to T R +, if and only if there is a Thorin function h such that λ = Λµ h, where µ h is in T R + with Thorin function and measure h and U h respectively. Since µ h and Λµ h have the same Lévy measure 8 ν µ hdx = dx x e xs U µ hds, from 2 and 7, the free cumulant transform of λ is obtained as dx 9 C λ z = xz e xs U x µ hdsdx = s G Esz U µ ds z C, which proves 6 and the if part of the second statement of theorem. For the converse, let U h be a Thorin measure and λ be a probability measure such that 6 is satisfied. Let ν µ hdx be the Lévy measure given by 8 and let µ h be the corresponding measure in T R +. Then, from 9 and the uniqueness of the Lévy-Khintchine representation, λ has Lévy measure ν µ hdx. Thus, by Bercovici-Para bijection λ = Λµ h and therefore λ T R +. Finally, to prove the first statement of the theorem, we use 7 in 9, proceed as in 7 and by using 5 we obtain that dx C λ z = xz e xs U x µ hdsdx = Thus, 4 and 6 are equivalent. hsg Ex hs z ds z C. 7

9 Using Propositions 2.2 and 2.3, we can easily deduce integral representations for FGGC. First, if Y h T R + has Wiener-Gamma representation 6, then ΛY h = L htdλγ t. Secondly, for any µ in I R, define the mapping Υ as Υ µ = L log t dzµ t. where Z µ t is free Lévy process with L Z µ = µ. Then it is easily seen that ΛΥ µ = Υ Λµ and that T R = Υ L R. Moreover, 2 T R { } = L e tdz µ t : µ ΛIlogR. We now consider some examples of FGGC. A probability measure µ on R is called free stable -stable, if the class {ψµ : ψ is an increasing affine transformation} is closed under the operation. Let S R denote the class of all free stable distributions on R. The free domains of attractions of S R were studied in [5]. As in the classical case, only the free Gaussian, the Cauchy and the free /2 stable have densities with closed form [5]. In the next example we further study the free /2 stable, pointing out that it is also infinitely divisible and GGC in the classical sense. Example 3.2. Let µ be the law of classical -stable law sometimes called Lévy distribution with scale parameter c and drift c so its Lévy measure is νdr = cr 3/2 dr. 2 It is easy to see that Λµ has density gx = c x c c2 4 x > c2 π x c c with Laplace transform E Λµ [exp rx] = 2 c 2 r π exp 4 + c t + 2 t 2 exp rc2 4 t dt r >. From this expression we deduce that Λµ is the Beta distribution of the second kind B 2, 3. Bondesson [8, pp 59] proved that Beta distributions of second kind are GGC. 2 2 Thus, Λµ belongs to T R + and T R +. It is an open problem whether free stable distributions other than free Cauchy and free -stable are also infinitely divisible in the 2 classical sense. Example 3.3. We compute the free cumulant transform of four FGGC examples arising from classical GGC whose Thorin measures are considered in []. From these expressions their corresponding free cumulants are readily obtained. 8

10 Let µ be in T R + with the Thorin measure U µ dx = n= δ π 2 dx. 8 2n 2 Then, 8 C µ z = π 2 2n G 2 E π2 8 2n 2 z z C n= { } 8 = k! z k+ z C. π 2 2n 2 k= 2 Let µ be in T R + with the Thorin measure U µ dx = C µ z = = k= n= 2 π 2 n G 2 E π2 n 2 z z C 2 n= k+ 2 k! π 2 n= n= δ π 2 n 2 2 z k+ z C. n 2k+ dx. Then, 3 Let µ be in T R + with the Thorin measure U µ dx = e xu,2 dx. Then, u2 u C µ z = k= x 2k e x 2 2k k! 2 z x dx z C. 4 Let µ be in T R + with the Thorin measure U µ dx = 2, dx Then, u2 u C µ z = e 2+sx ds ss + 2 z x dx. 4. Random Matrix Models for Free GGC Let M d = M d C denote the linear subspace of Hermitian matrices, with scalar product A, B trab, for A, B M d and tr denotes trace. By M we denote the Euclidean norm. Let M + d be the closed cone of nonnegative definite matrices in M d. Let us first recall several facts on infinite divisibility of matrices taking values in the cone M + d see [2]. A d d Hermitian random matrix M is infinitely divisible in M+ d if and only if its cumulant transform CM A = log E[expiT ram] is of the form 2 CMA = itrθ A + e itrxa ρdx, A M + m, M + d where Θ M + d is called the drift and the Lévy measure ρ is such that ρm d\m + d = and ρ has order of singularity 22 min, X ρdx <. Moreover, the Laplace transform of M is given by { 23 E[exp trm A] = exp trθ A M + d M + d e trxa ρdx }. 9

11 If M is an infinitely divisible matrix in M + d, the associated matrix Lévy process {M t} t is called a matrix subordinator. It is M + d -increasing in the sense that for all s < t, M t M s M + d with probability one. The matrix valued random integral 24 N = ftdm t of a non-random real valued function f is defined in the sense of integrals with respect to scattered random measures, see [2], [4]. When definable, it is a d d infinitely divisible random matrix with cumulant transform 25 C NA = C MftAdt. Of special interest in this work is the Gamma type matrix subordinator Γ = {Γ d t } t corresponding to the Lévy measure 26 ρ g d exp X dx = ω d dx X where ω d /de = dr S d ω d dv E rv. ω d is the probability measure on S + d = {A M + d ; A = } induced by the transformation u V = uu, where the column random vector u is uniformly distributed on the unit sphere of C d. The Lévy measure has the polar decomposition ρ g d 27 ρ g d E = d S + d ω d dv E rv e r r dr. We observe that ρ g d has support on the subset of rank one matrices in M+ d. The case d = corresponds to the Lévy measure of the one dimensional gamma process. The corresponding matrix random integral htdγ d t is called the matrix Wiener-Gamma integral and is defined for Borel functions h : R + R + satisfying 5. The following is the main result of this section. It gives a RMM for FGGC on R +, where the RMM is given by matrix Wiener-Gamma type integrals, which are GGC matrix extensions of the one-dimensional case. Theorem 4.. Let µ h be a classical GGC on R + given by the Wiener-Gamma integral µ h = L htdγ t. The free GGC Λµ h has a RMM given by the ensemble of infinitely divisible matrix Wiener-Gamma integrals 28 Mh d = htdγ d t, d where for each d, {Γ d t } t is the Gamma type matrix subordinator associated to the Lévy measure ρ g d given by 26.

12 Proof. We shall use Theorem 6. in [4], which establishes that for any µ I R, there is an ensemble of random matrices M d d such that the spectral distribution of M d converges in probability to Λµ. Moreover, from Theorem 3. in [4], for each d, the Fourier transform of the random matrix M d is given by the expression 29 E[expiTrAM d ] = exp{e u d C µ u, Au }, A M m, where u = u,..., u d t is a uniformly distributed random vector on the unit sphere of C d and Cµ is the cumulant function Lévy exponent of µ. We will show that when µ h is a classical GGC, the random matrices Mh d d given by 28 have the same laws as M d d with Fourier transform 29, where Cµ is the cumulant transform C of µ h. This µ h will prove the theorem. First, let µ be the one dimensional standard Gamma distribution, d be fixed and u = u,..., u d t be a uniformly distributed random vector on the unit sphere of C d. Let Γ d be the Gamma type matrix subordinator at t = corresponding to the Lévy measure 26. We will show that [ 3 E[exp TrAΓ d ] = exp { de ]} u e u,au x e x x dx. Then, writing V = uu and using the polar decomposition 27 we have { E[exp TrAΓ d ] = exp } e T rax e X M + X ω ddx d { = exp d ω d dv } e T rv Ax e x S + d x dx [ = exp { de ]} V e T rv Ax e x x dx [ = exp { de ]} u e T ruu Ax e x 3 x dx. Second, let P µh d d be the matrix distributions of the random matrices ensemble given by 28, where µ h is a classical one dimensional GGC with Thorin function h. Using 25, 3 and 27, we have that E P µ [exp TrAMh] d = exp h d { { ds } e T raxhs e X M + X ω ddx d exp d ds ω d dv } e T rv Ahsx e x S + d x dx = exp { de u ds } e T ruu Ax e x/hs dx. x From this Laplace transform and 8, we get 29.

13 Remark 4.2. If µ T R + and without drift, then Λµ is concentrated on R +. This follows trivially from the above construction of the RMM. As pointed out by the referee, this fact also follows from the well known equivalence ν n n n µ ν n n n Λµ. Similar to the above theorem, we can construct RMM for GGC on R, where the RMM is given by matrix random integrals similar to the one dimensional representation. Theorem 4.3. Let µ be in T R given by the random integral representation µ = L e tdx µ t, for µ Ilog R and where Xµ t is a Lévy process such that LX µ = µ. The free GGC Λµ has a RMM given by the ensemble of infinitely divisible matrix random integrals 32 Mh d = e tdrt d, d where for each d, {Rt d } t is a matrix valued Lévy process with Lévy measure ν d given by ν d E = ω d dv E rv νdr, with ω d as in 26 and ν is the Lévy measure of µ. S + d 5. Inheritance of nested subclasses of FGGC and its limit class under Λ Maejima and Sato [] proved that nested subclasses of classical Thorin distributions are characterized by limit theorem and proved that its limit class is the closure of the class of classic stable distributions S R, which is taken under -convolution and weak convergence. We now point out a similar result for free Thorin distributions. The free selfdecomposable case was recently considered by Sakuma [5]. We define subclasses of T R as follows. Let Ψ = e tdz µ t be the free integral considered in 2 and Ilog mr = {µ I R : R log+ x m µdx < }. For m =, 2,...let TḿR = ΛΨI log R and T R = m=tḿr. m+ 2 µ LḿR if, for any c,, there exists ρ c Lḿ R such that µ = D c µ ρ c. We also define L R = m=lḿr. It was proved in [5] that LḿR is -c.c.s.s. and L = S R. The following concept was introduced in the sense of classical convolution in []. Definition 5.. A class M of distributions on R is said to be resp. -completely closed in the strong sense -c.c.s.s. resp. -c.c.s.s., if M I R resp. M I R and if the following are satisfied. It is closed under resp. -convolution. 2 It is closed under weak convergence. 3 If µ M, then D c µ δ b M resp. D c µ δ b M for any c > and b R. 4 µ M implies µ s M resp. µ s M for any s >, where µ s is the distribution with the cumulant sc µ z resp. µ s is the distribution with the free cumulant sc µ z. 2

14 The closure is taken under -convolution and weak convergence. The following result gives the preservation of classical completely closed in the strong sense class under the Bercovici-Pata bijection. Lemma 5.2. If M is -c.c.s.s., then ΛM is -c.c.s.s.. Proof. and 2 in the above definition follow from Proposition 2.2. If µ ΛM, then Λ D c µ δ b = D c Λ µδ b M. So D c µ δ b ΛM and 3 holds. Finally, 4 holds from the classical and free Lévy Khintchine formulas. From the above lemma and Proposition 2.3, we immediately obtain the following relationships. Lemma 5.3. Fix < a <. Suppose f is continuous on, a and a fsds. Let {Z t } be a free Lévy process with distribution µ. Define the mapping a Φ f µ = L fsdz s µ. Then the following are true If M is -c.c.s.s., then Φ f M M. 2 If M is -c.c.s.s., then Φ f M is also -c.c.s.s. Theorem 5.4. T R = L R = S R. Proof. From TḿR = Υ m+ LḿR LḿR, we have 33 T R L R. Since TmR is -c.c.s.s., then TḿR is -c.c.s.s. It is clear that TḿR = Υ m+ LḿR Υ m+ S R = S R. Next, since TḿR is -c.c.s.s., TḿR S R and therefore, 34 T S R = L R. Then 33 and 34 yield Acknowledgment. T R = S R = L R. The authors would like to thank the referee for a very careful reading of the manuscript and for the valuable suggestions and comments. References [] O. E. Barndorff-Nielsen, M. Maejima and K. Sato 26. Some classes of multivariate infinitely divisible distributions admitting stochastic integral representation, Bernoulli [2] O. E. Barndorff-Nielsen and V. Pérez-Abreu 28. Matrix subordinators and related Upsilon transformations, Theory Probab. Appl. 52, -27. [3] O. E. Barndorff-Nielsen and S. Thorbjørnsen 26. Classical and free infinite divisibility and Lévy processes, Lecture Notes in Math. 866, Springer,

15 [4] F. Benaych-Georges 25. Classical and free infinitely divisible distributions and random matrices, Ann. Probab. 33, [5] H. Bercovici and V. Pata with an appendix by P. Biane 999. Stable laws and domains of attraction in free probability theory, Ann. Math. 49, [6] H. Bercovici and D. Voiculescu 993. Free convolution of measures with unbounded support, Indiana J. Math. 42, [7] A. Lijoi and E. Regazzini 24. Means of a Dirichlet process and multiple hypergeometric functions, Ann. Probab. 32, [8] L. Bondesson 992. Generalized gamma convolutions and related classes of distribution and densities, Lecture Notes in Statist. 76, Springer-Verlag, New York. [9] T. Cabanal-Duvillard 25. A matrix representation of Bercovici-Pata bijection, Elect. J. Probab., [] L. F. James, B. Roynette and M. Yor 27. Generalized gamma convolutions, Dirichlet means, Thorin measures, with explicit examples. arxiv: v [math.pr] 29 Aug 27. [] M. Maejima and K. Sato 28. The Limits of nested subclasses of several classes of infinitely divisible distributions are identical with the closure of the class of stable distributions, Probab. Theory Related Fields, To appear.. [2] B. Rajput and J. Rosinski 989. Spectral representation of infinitely divisible processes, Probab. Theory Related Fields 82, [3] K. Sato 999. Lévy Processes and Infinitely Divisible Distributions, Cambridge University Press, Cambridge. [4] K. Sato 26. Additive processes and stochastic integrals, Illinois J. Math. 5, [5] N. Sakuma 28. Characterizations of the class of free self decomposable distributions and its subclasses, Inf. Dim. Anal. Quantum Probab. To appear. [6] O. Thorin 977a. On the infinite divisibility of the Pareto distribution, Scand. Actuar. J., 3, 3-4. [7] O. Thorin 977b. On the infinite divisibility of the lognormal distribution, Scand. Actuar. J., 3, [8] O. Thorin 978. An extension of the notion of a generalized Γ-convolution, Scand. Actuar. J., 4, Department of Probability and Statistics, CIMAT, Apdo. Postal 42, Guanajuato, Gto. México, pabreu@cimat.mx Department of Mathematics, Keio University, 3-4- Hiyoshi, Kohoku-ku, Yokohama, Japan, noriyosi@math.keio.ac.jp 4