FREE GENERALIZED GAMMA CONVOLUTIONS

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 * ( + ) of classical infinitely divisible distributions on + that contains all Gamma distributions and that is closed under classical convolution and weak convergence. This class was introduced by Thorin [16], [17] and further studied by Bondesson [7]. Thorin [18] also considered the smallest class of distributions on the real line which contains all distributions in T * ( + ) and is closed under convolution, convergence and reflection. We denote this class by T * ( ) and called it the Thorin class of distributions on . Let ( ) be set of probability measures on and I * ( ) the class of all classical infinitely divisible distributions in ( ). If µ ∈ ( ),μ(z) denotes its Fourier transform and when µ ∈ I * ( ) 1  we denote by * µ (z) its classical cumulant function or Lévy exponent i.e. * µ (z) = logμ(z). A probability measure µ ∈ ( ) is in I * ( ) if and only if its classical cumulant function has the Lévy-Khintchine representation : * where a ≥ 0, γ ∈ and ν (the so called Lévy measure) is a measure satisfying ν({0}) = 0 and (1∧|x| 2 ) < ∞. The triplet (a, η, ν) is uniquely determined and is called * -characteristic triplet or simply * -triplet. When |x|ν(d x) < ∞, we speak of the drift type Lévy Khintchine representation * where η ′ is the drift of µ and is given by η ′ = η − {|x|≤1} xν(d x). We write I * log ( ) = µ ∈ I * ( ); log(|x| ∧ 1)µ(dx) < ∞ and refer to Sato [13] for basic facts about classical infinitely divisible distributions. Bondesson [7] proved that a positive random variable Y with classical law * (Y ) = µ -without translation term-belongs to T * ( + ) if and only if there exists a positive Radon measure U µ on (0, ∞) such that * x < ∞. The measure U µ is called the Thorin measure of µ. So, the * -triplet of µ is (0, 0, ν µ ) where the Lévy measure is concentrated on (0, ∞) and such that It is known that the class T * ( + ) is characterized by Wiener-Gamma representations, i.e., random integral representations with respect to the standard one-dimensional Gamma process (see [10], [9]). Specifically, a positive random variable Y belongs to T * ( + ) if and only if there is a Borel function h : where γ t ; t ≥ 0 is the standard Gamma process with Lévy measure where U h µ denotes the image of Lebesgue measure on (0, ∞) under the application : The function h is called the Thorin function of Y and is obtained as follows.
Moreover we have the following alternative expression for the cumulant function of µ * In the above equation to indicate that it has the integral representation (6) and write µ h = * (Y h ). We have excluded from the above discussion the case of non-zero drift, which is easily incorporated by considering nonzero drift c 0 in the * -triplet (c 0 , 0, ν µ ).
Many well known distributions belong to T * ( + ). The positive α-stable distributions, 0 < α < 1, are GGC with h(s) = {sθ Γ(α + 1)} − 1 α for a θ > 0. In particular, for the 1/2−stable distribution, h(s) = 4 s 2 π −1 . First passage time distribution, Beta distribution of the second kind, lognormal and Pareto are also GGC, see [9]. As for distributions in T * ( ), there is a another random integral representation approach recently presented in Barndorff-Nielsen et. al [1], who also considered the multivariate case. We recall that if (X t ; t ≥ 0) is a * -Lévy process and f : [a, b] → is a continuous function defined on an interval [a, b] in [0, ∞), then the stochastic integral [a,b] f (t)d X t may be defined as the limit in probability of approximating Riemann sums. Moreover, if f is continuous function defined on [0, ∞), [a,∞) f (t)d X t may be as the limit in probability of [a,b) f (t)d X t when b → ∞. For stochastic integrals of nonrandom functions with respect to general additive processes we refer to Sato [14]. It is shown in [1], that for any µ in I * ( ), the mapping Υ * given by is always defined, where X (µ) t is a Lévy process with * (X (µ) where the function e −1 1 (t) is the inverse of the incomplete gamma function e 1 (x) = ∞ x e −s s −1 ds and X (µ) t is a Lévy process with * (X (µ) In the study of relations between classical and free infinite divisibility, Bercovici and Pata [5] introduced a bijection Λ between the set I * ( ) of classical infinitely divisible laws and the set I ⊞ ( ) of free infinitely divisible laws. A new approach to this bijection was recently proposed by Benaych-Georges [4] and Cabanal-Duvillard [8]. 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≥1 where for each d ≥ 1, M d is a d×d matrix with random entries. The (random) spectral measure (or empirical spectral law) 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], [8] that for any µ ∈ I * ( ), there exists a random matrix model (M d ) d≥1 for Λ(µ), 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 (10), 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 Λ(T * ( )) and their limits, analogous to the recent results for the classical convolution case study in Maejima and Sato [11].
(2) Let δ a be Dirac measure at a. Λ(δ a ) = δ a for a ∈ . So Λ is preserved under affine transforms, i.e. Λ(D c µ * δ a ) = D c Λ(µ) ⊞ δ a for any b > 0 and a ∈ where D c µ means the spectral distribution of the operator cX with µ = (X ).
For a classical random variable X or a stochastic process (X t ), we write Λ(X ) and Λ(X t ) as a short notation for Λ( * (X )) and Λ( * (X t )).

Barndorff-Nielsen and Thorbjørnsen
We denote by L ⊞ ( ) the class of all free selfdecomposable distributions on . We refer to Sakuma [15] 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 ⊞ (Z) its spectral distribution. Following [3], we say that a process (Z t ; t ≥ 0) of selfadjoint operators affiliated with a W * -probability space ( , τ), is a free Lévy process (in law) if it satisfies the following four conditions: (1) Z 0 = 0 (2) Whenever n ∈ and 0 ≤ t 0 < t 1 < · · · t n , the increments are freely independent random variables.
In particular, if Y is a free selfdecomposable random variable, there exists a free Lévy process Z t such that

Free Generalized Gamma Convolutions
When γ is the classical gamma distribution, we call Λ(γ) the free gamma distribution. If (γ t ; t ≥ 0) is the standard Gamma process, the free Lévy process (Λ(γ t ); t ≥ 0) 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 = Λ(µ). We denote by T ⊞ ( + ) = Λ(T * ( + )) and T ⊞ ( ) = Λ(T * ( )) the classes of FGGC and Free Thorin class respectively. It follows trivially from Proposition 2.2, that T ⊞ ( + ) is the smallest class that contains all free Gamma distributions and that is closed under ⊞-convolution and convergence, while T ⊞ ( ) is the smallest class on the real line which contains T ⊞ ( + ) and is closed under convolution, convergence and reflection.
The following result is a characterization of the free cumulant transform of distributions in T ⊞ ( + ) in terms of the Cauchy transform of the exponential distribution.
where G E(a) is the Cauchy transform of the exponential law with mean 1/a, i.e.
Alternatively, a probability measure λ in + is FGGC without drift term if and only if there is a Thorin Proof. For any t ≥ 0, the Lévy measure of (γ t ) has finite first moment. We work with the drift type representation (12) with η ′ µ = a µ = 0. First, since (γ t ) and (Λ(γ t )) have the same characteristic * and ⊞-triplet, from (12), the free cumulant transform of Λ(γ t ) is obtained as Next, by Remark 1.1, a probability measure λ without drift term belongs to T ⊞ ( + ), if and only if there is a Thorin function h such that λ = Λ(µ h ), where µ h is in T * ( + ) with Thorin function and measure h and U h respectively. Since µ h and Λ(µ h ) have the same Lévy measure from (12) and (17), the free cumulant transform of λ is obtained as which proves (16) 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 (16) is satisfied. Let ν µ h (dx) be the Lévy measure given by (18) and let µ h be the corresponding measure in T * ( + ). Then, from (19) and the uniqueness of the Lévy-Khintchine representation, λ has Lévy measure ν µ h (dx). Thus, by Bercovici-Para bijection λ = Λ(µ h ) and therefore λ ∈ T ⊞ ( + ). Finally, to prove the first statement of the theorem, we use (7) in (19), proceed as in (17) and by using (15) we obtain that Thus, (14) and (16) Secondly, for any µ in I ⊞ ( ), define the mapping Υ ⊞ as where Z µ t is free Lévy process with ⊞ (Z (µ) Then it is easily seen that Λ(Υ * (µ)) = Υ ⊞ (Λ(µ)) and that T ⊞ ( ) = Υ ⊞ (L ⊞ ( )). Moreover, We now consider some examples of FGGC. A probability measure µ on is called free stable (⊞-stable), if the class {ψ(µ) : ψ is an increasing affine transformation} is closed under the operation ⊞. Let S ⊞ ( ) denote the class of all free stable distributions on . The free domains of attractions of S ⊞ ( ) were studied in [5]. As in the classical case, only the free Gaussian, the Cauchy and the free 1/2−stable have densities with closed form [5]. In the next example we further study the free 1/2−stable, pointing out that it is also infinitely divisible and GGC in the classical sense. 1 2 -stable law (sometimes called Lévy distribution) with scale parameter c and drift c 0 ≥ 0 (so its Lévy measure is ν(dr) = cr −3/2 dr). It is easy to see that Λ(µ) has density

Example 3.2. Let µ be the law of classical
From this expression we deduce that Λ(µ) is the Beta distribution of the second kind B 2 ( 1 2 , 3 2 ). Bondesson [7, pp 59] proved that Beta distributions of second kind are GGC. Thus, Λ(µ) belongs to T * ( + ) and T ⊞ ( + ). It is an open problem whether free stable distributions other than free Cauchy and free 1 2 -stable are also infinitely divisible in the 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 [9]. From these expressions their corresponding free cumulants are readily obtained.

Random Matrix Models for Free GGC
Let d = d ( ) denote the linear subspace of Hermitian matrices, with scalar product 〈A, B〉 → tr(AB * ), for A, B ∈ d and tr denotes trace. By M we denote the Euclidean norm. Let + d be the closed cone of nonnegative definite matrices in d . Let us first recall several facts on infinite divisibility of matrices taking values in the cone + d (see [2]). A d × d Hermitian random matrix M is infinitely divisible in + d if and only if its cumulant transform * M (A) = log [exp(iTr(AM ))] is of the form * where Θ 0 ∈ + d is called the drift and the Lévy measure ρ is such that ρ( d \ + d ) = 0 and ρ has order of singularity Moreover, the Laplace transform of M is given by If M is an infinitely divisible matrix in + d , the associated matrix Lévy process {M t } t≥0 is called a matrix subordinator. It is + d -increasing in the sense that for all 0 ≤ s < t, M t − M s ∈ + d with probability one. The matrix valued random integral of a non-random real valued function f is defined in the sense of integrals with respect to scattered random measures, see [12], [14]. When definable, it is a d × d infinitely divisible random matrix with cumulant transform * Of special interest in this work is the Gamma type matrix subordinator Γ = {Γ d t } t≥0 corresponding to the Lévy measure where where the column random vector u is uniformly distributed on the unit sphere of d . The Lévy measure ρ g d has the polar decomposition We observe that ρ g d has support on the subset of rank one matrices in + d . The case d = 1 corresponds to the Lévy measure of the one dimensional gamma process. The corresponding matrix random integral ∞ 0 h(t)dΓ d t is called the matrix Wiener-Gamma integral and is defined for Borel functions h : + → + satisfying (5). The following is the main result of this section. It gives a RMM for FGGC on + , where the RMM is given by matrix Wiener-Gamma type integrals, which are GGC matrix extensions of the one-dimensional case.

The free GGC Λ(µ h ) has a RMM given by the ensemble of infinitely divisible matrix Wiener-Gamma integrals
where for each d ≥ 1, {Γ d t } t≥0 is the Gamma type matrix subordinator associated to the Lévy measure ρ g d given by (26). Proof. We shall use Theorem 6.1 in [4], which establishes that for any µ ∈ I * ( ), there is an ensemble of random matrices (M d ) d≥1 such that the spectral distribution of M d converges in probability to Λ(µ). Moreover, from Theorem 3.1 in [4], for each d ≥ 1, the Fourier transform of the random matrix M d is given by the expression where u = (u 1 , ..., u d ) t is a uniformly distributed random vector on the unit sphere of d and * µ is the cumulant function (Lévy exponent) of µ. We will show that when µ h is a classical GGC, the random matrices (M d h ) d≥1 given by (28) have the same laws as (M d ) d≥1 with Fourier transform (29), where * µ is the cumulant transform * µ h of µ h . This will prove the theorem. First, let µ be the one dimensional standard Gamma distribution, d ≥ 1 be fixed and u = (u 1 , ..., u d ) t be a uniformly distributed random vector on the unit sphere of d . Let Γ d 1 be the Gamma type matrix subordinator at t = 1 corresponding to the Lévy measure (26). We will show that Then, writing V = uu * and using the polar decomposition (27) we have From this Laplace transform and (8), we get (29).

Remark 4.2.
If µ ∈ T * ( + ) and without drift, then Λ(µ) is concentrated on + . 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→∞ Λ(µ).
Similar to the above theorem, we can construct RMM for GGC on , where the RMM is given by matrix random integrals similar to the one dimensional representation (10).

Theorem 4.3.
Let µ 1 be in T * ( ) given by the random integral representation for µ ∈ I * log ( ) and where X (µ) t is a Lévy process such that (X (µ) 1 ) = µ. The free GGC Λ(µ 1 ) has a RMM given by the ensemble of infinitely divisible matrix random integrals where for each d ≥ 1, {R d t } t≥0 is a matrix valued Lévy process with Lévy measure ν d given by with ω d as in (26) and ν is the Lévy measure of µ.

Inheritance of nested subclasses of FGGC and its limit class under Λ
Maejima and Sato [11] 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 * ( ), 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 [15]. We define subclasses of T ⊞ ( ) as follows. Let Ψ = (2) µ ∈ L ⊞ m ( ) if, for any c ∈ (0, 1), there exists ρ c ∈ L ⊞ m−1 ( ) such that µ = D c µ ⊞ ρ c . We also define L ⊞ ∞ ( ) = ∩ ∞ m=0 L ⊞ m ( ). It was proved in [15] that L ⊞ m ( ) is ⊞-c.c.s.s. and L ⊞ ∞ = S ⊞ ( ). The following concept was introduced in the sense of classical convolution in [11]. (4) µ ∈ M implies µ s * ∈ M (resp. µ s⊞ ∈ M) for any s > 0, where µ s * is the distribution with the cumulant sC µ (z) (resp. µ s⊞ is the distribution with the free cumulant s ⊞ µ (z)). 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. From the above lemma and Proposition 2.3, we immediately obtain the following relationships. Proof Since T * m ( ) is * -c.c.s.s., then T ⊞ m ( ) is ⊞-c.c.s.s. It is clear that T ⊞ m ( ) = Υ m+1 ⊞ (L ⊞ m ( )) ⊂ Υ m+1 ⊞ (S ⊞ ( )) = S ⊞ ( ). Next, since T ⊞ m ( ) is ⊞-c.c.s.s., T ⊞ m ( ) ⊂ S ⊞ ( ) and therefore, Then (33) and (34) yield T ⊞ ∞ ( ) = S ⊞ ( ) = L ⊞ ∞ ( ).