Free infinite divisibility for beta distributions and related ones

We prove that many of beta, beta prime, gamma, inverse gamma, Student t- and ultraspherical distributions are freely infinitely divisible, but some of them are not. The latter negative result follows from a local property of probability density functions. Moreover, we show that the Gaussian, ultraspherical and many of Student t-distributions have free divisibility indicator 1.


Beta and beta prime distributions
Wigner's semicircle law w and a Marchenko-Pastur law (or a free Poisson law) m, defined by are the most important distributions in free probability because they are respectively the limit distributions of free central limit theorem and free Poisson's law of small numbers. In the context of random matrices, w and m are the large N limit of the eigenvalue distributions of X N and X 2 N respectively, where X N is an N × N normalized Wigner matrix. Those measures belong to the class of freely infinitely divisible (or FID for short) distributions, the main subject of this paper. This class appears as the spectral distributions of large random matrices [BG05,C05]. Research on free probability or more specifically FID distributions has motivated some new directions in classical probability: upsilon transformation (see [BT06]), the class of type A distributions [ABP09,MPS12] and matrix-valued Lévy processes [AM12]. Handa [H12] found a connection of branching processes and GGCs to Boolean convolution (see Section 7), a convolution related to free probability.
Up to affine transformations, w and m are special cases of beta distributions: where B(p, q) is the beta function 1 0 x p−1 (1 − x) q−1 dx. Moreover, β 1/2,1/2 is an arcsine law which appears in monotone central limit theorem [M01] and plays a central role in free type A distributions [ABP09]. If p = q, the beta distribution β p,p can be shifted to a symmetric measure which is called the ultraspherical distribution essentially. This family contains Wigner's semicircle law and a symmetric arcsine law. If we take a limit p → 0, a Bernoulli law appears, which is known as the limit distribution of Boolean central limit theorem [SW97]. In the case p + q = 2, the measure β p,q has explicit Cauchy and Voiculescu transforms [AHb]. Moreover, if we let β a := β 1−a,1+a , −1 < a < 1, it holds that (D b β a ) ⊲ β b = β ab , a, b ∈ (−1, 1). (1.1) The binary operation ⊲ is monotone convolution [M00, F09] and D a µ is the dilation of a probability measure µ by a: (D a µ)(A) := µ( 1 a A) for Borel sets A ⊂ R and a = 0. D 0 µ is defined to be δ 0 .
Beta prime distributions 1 β ′ p,q (dx) := 1 B(p, q) x p−1 (1 + x) p+q 1 [0,∞) (x) dx, p, q > 0, also appear related to free probability. The measure β ′ 3/2,1/2 is a one-sided free stable law with stability index 1/2; see p. 1054 of [BP99]. The same measure also appears as the law of an affine transformation of X −1 when X follows the free Poisson law m. If X follows the semicircle law w, then 1 X+2 follows the beta prime distribution β ′ 3/2,3/2 up to an affine transformation. If X follows a Cauchy distribution, i.e. a free stable law with stability index 1, then X 2 follows the beta prime distribution β ′ 1/2,1/2 . Thus various beta and beta prime distributions appear in noncommutative probability. We will investigate free infinite divisibility for these distributions.
(2) lim q→∞ D q β p,q = γ p in the sense of weak convergence.
The measures β ′ p,q , γ p , γ −1 p , t q are all infinitely divisible in the classical sense for all possible parameters p, q; see [B92], p. 59 and p. 117. Combining the main theorem and its corollary below, we have many probability measures which are both freely and classically infinitely divisible.

Main results
The aim of this paper is to understand free infinite divisibility of beta, beta prime and related distributions. First we summarize the known results. It is well known that Wigner's semicircle law and the free Poisson law are FID. β a = β 1−a,1+a is FID if (and only if) 1 2 ≤ |a| < 1 [AHb]. The free infinite divisibility for ultraspherical distributions u p was conjectured for p ≥ 1 in [AP10,Remark 4.4], and Arizmendi and Belinschi [AB] showed that the ultraspherical distribution u n (and also the beta distribution β 1 2 ,n+ 1 2 ) is FID for n = 1, 2, 3, · · · . For beta prime distributions, β ′ 2/3,1/2 is a free stable law and so is FID [BP99, p. 1054]. β ′ 1/2,1/2 is also known to be FID because it is the square of a Cauchy distribution [AHS]. The t-distribution t q is FID for q = 1, 2, 3, · · · [H]. The chi-square distribution 1 √ πx e −x 1 [0,∞) (x) dx coincides with γ 1/2 and it is FID [AHS], while the exponential distribution is not FID. 2 The main theorem of this paper is the following, which is proved through Sections 3-6.
(1) The beta distribution β p,q is FID in the following cases: (2) The beta distribution β p,q is not FID in the following cases: (i) 0 < p, q ≤ 1; (ii) p ∈ I; (iii) q ∈ I, where The assertions (2) and (4) follow from Theorem 5.1, a general criterion for a probability measure not to be FID. It roughly says, if a probability measure has a local density function p(x) around a point x 0 , and if p(x)| (x 0 −δ,x 0 +δ) is close to the power function for some c, δ > 0 and α ∈ I, then that measure is not FID.
Theorem 1.1 has the following consequences.
, and is not FID if p ∈ I.
(2) The inverse gamma distribution γ −1 p is FID for any p > 0. In particular, the classical positive stable law with stability index 1/2 is FID.
This paper is organized as follows. General results on FID distributions are developed in Section 2. The main theorem is proved in Sections 3, 5 and 6.
In Section 7, we will provide a method for computing the free divisibility indicator of a symmetric measure and show that ultraspherical distributions and t-distributions mostly have free divisibility indicators equal to 1. Also the Gaussian distribution has the value 1.
In the final section, we gather explicit Cauchy transforms of beta and beta prime distributions. The measures β a := β a,1−a and β ′ a (dx) := β ′ 1−a,a (dx − 1) are shown to satisfy 2 Free infinite divisibility 2.1 Preliminaries 1. Tools from complex analysis. Let C + , C − , H + and H − be the upper half-plane, lower half-plane, right half-plane and left half-plane, respectively. Given a Borel probability measure µ on R, let G µ be its Cauchy transform defined by is called the reciprocal Cauchy transform of µ. When the Cauchy For a random variable X ∼ µ, we may write G X , G X instead of G µ , G µ respectively. A measure µ can be recovered from G µ or G µ by using the Stieltjes inversion formula [A65,Page 124]: for all continuity points a, b of µ. In particular, if the functions f y µ (x) := − 1 π Im G µ (x + iy) converge uniformly to a continuous function f µ (x) as y ց 0 on an interval [a, b], then µ is absolutely continuous on [a, b] with density f µ (x). Atoms can be identified by the formula µ({x}) = lim yց0 iyG µ (x + iy) for any x ∈ R.
(1) The reciprocal Cauchy transform F µ is an analytic map of C + to C + .
(2) F µ satisfies Im F µ (z) ≥ Im z for z ∈ C + . If there exists z ∈ C + such that Im F µ (z) = Im z, then µ must be a delta measure δ a .
(3) For any ε, In addition, the following property is used in Section 6.
Note that some symmetric probability measures do not satisfy the property Re 2. Free convolution and freely infinitely divisible distributions. If X 1 , X 2 are free random variables following probability distributions µ 1 , µ 2 respectively, then the probability distribution of X 1 + X 2 is denoted by µ 1 ⊞ µ 2 and is called the free additive convolution of µ 1 and µ 2 . Free additive convolution is characterized as follows [BV93]. From Proposition 2.1(4), for any λ > 0, there is M > 0 such that the right compositional inverse map F −1 µ exists in Γ λ,M . Let φ µ (z) be the Voiculescu transform of µ defined by The free convolution µ ⊞ ν is the unique probability measure such that in a common domain of the form Γ λ ′ ,M ′ . Free convolution associates a basic class of probability measures, called freely infinitely divisible distributions introduced in [V86] for compactly supported probability measures and in [BV93] for all probability measures.
n times The set of FID distributions is closed with respect to the weak convergence [BT06,Theorem 5.13]. FID distributions appear as the limits of infinitesimal arrays as in classical probability theory; see [CG08].
FID distributions are characterized in terms of a complex analytic property of Voiculescu transforms.
Theorem 2.4 ( [BV93]). For a probability measure µ on R, the following are equivalent.
Note that Pick functions are also crucial in the characterization of generalized gamma convolutions (GGCs) in classical probability [B92].

Sufficient conditions for free infinite divisibility
When the Voiculescu transform does not have an explicit expression, the conditions in Theorem 2.4 are difficult to check. In such a case, a subclass U I of FID measures has been exploited in the literature [BBLS11,ABBL10,AB,AHa,AHb,BH,H]. We also introduce a variant of it.
Definition 2.5. (1) A probability measure µ is said to be in class U I if F −1 µ , defined in some Γ λ,M , analytically extends to a univalent map in C + . µ ∈ U I if and only if there is an open set Ω ⊂ C, Ω ∩ Γ λ,M = ∅ such that F µ extends to an analytic bijection of Ω onto C + .
(2) A symmetric probability measure µ is said to be in class U I s if: (a) there is c ≤ 0 such that F µ extends to a univalent map around i(c, ∞) and maps i(c, ∞) onto i(0, ∞); (b) there is an open set Ω ⊂ C − ∪ H + such that Ω ∩ Γ λ,M = ∅ for some λ, M > 0 and that F µ extends to an analytic bijection of Ω onto C + ∩ H + .
Remark 2.6. In [AHb] we required F µ to be univalent in C + in the definition of µ ∈ U I, but this automatically follows. If F −1 µ is analytic in C + , then F −1 µ • F µ (z) = z for z ∈ C + by analyticity, so that F µ should be univalent in C + .
Lemma 2.7. If µ ∈ U I or µ ∈ U I s , then µ is FID.
Proof. The proof for U I is found in [AHb,BBLS11]. Assume µ ∈ U I s . We are able to define where Ω * := {−x + iy : x + iy ∈ Ω} and F µ | A is the restriction of F µ to a set A. This is well defined because each of Ω, i(c, ∞) and Ω * has nonempty intersection with Γ λ,M , and so each of coincides with the original inverse (2.2) in the common domain. Note that, as explained in Remark 2.6, F µ is univalent in C + .
The remaining proof is similar to the case µ ∈ U I. Take z ∈ C + ∩ H + . If z ∈ F µ (C + ), then taking the pre image w ∈ C + of z and we see Im The other two cases z ∈ i(0, ∞) and z ∈ C + ∩ H − are similar.
The following conditions on a Cauchy transform are quite useful to prove the free infinite divisibility of a probability measure.
Condition (A2) is useful to define an inverse map F −1 µ in C + . This condition was crucial in the proof of free infinite divisibility of Gaussian [BBLS11]. Condition (A3) is used to show the map F −1 µ is univalent in C + . (A3) is important as well as (A1) and (A2) because the exponential distribution satisfies (A1) and (A2) for D = C \ (−∞, 0], but does not satisfy (A3). It is known that the exponential distribution is not FID.
For symmetric distributions, the following variant can be more useful.
(1) If the Cauchy transform G µ of a probability measure µ satisfies (A), then µ ∈ U I.
(2) If the Cauchy transform of a symmetric probability measure µ satisfies (B), then µ ∈ U I s . If, moreover, the domain D can be taken as a subset of H + , then µ ∈ U I.

Proof.
(1) Let c t ⊂ C + be a curve defined by Note that t>0 c t = C + . From Proposition 2.1(2), for each t > 0, if we take a large R > 0, there exists a simple curve γ R t such that F µ (γ R t ) = c t ∩ {z ∈ C + : Re z > R} and F µ maps a neighborhood of γ R t onto a neighborhood of c t ∩ {z ∈ C + : Re z > R} bijectively. Take a sequence z n ∈ γ R t converging to the edge of γ R t which we denote by z R , then The following cases are possible: (i) When z converges to z 0 , the pre images w have an accumulative point w 0 in D; (ii) When z converges to z 0 , the pre images w have an accumulative point w 1 in ∂D ∪ {∞}.
In the case (i), we can still extend the curve γ t more because of condition (A2) and the obvious fact F µ (w 0 ) = z 0 ; a contradiction to the maximality of γ t . The point w 0 might be a pole of F µ , but in that case z 0 has to be infinity, which is again a contradiction. In the case (ii), analytically, and hence µ ∈ U I.
(2) The proof is quite similar. Let c t := c t ∩ H + . One can prolong the above γ R t , to obtain Remark 2.10. Condition (A2) enables us to construct the curve γ t , but γ t can enter another Riemannian sheet of F µ beyond ∂D. Condition (A3) becomes a "barrier" which prevents such a phenomenon. If F µ is meromorphic in C as in the case of the normal law, there is no other branch of F µ and so condition (A3) is not needed.

Cauchy transforms of beta, beta prime and Student t-distributions
Let F (a, b; c; z) be the Gauss hypergeometric series: with the conventional notation (a) n := a(a+1) · · · (a+n−1), (a) 0 := 1. This series is absolutely convergent for |z| < 1. There is an integral representation which continues F (a, b; c; z) analytically to C \ [1, ∞). The normalizing constant B(p, q) is the beta function which is related to the gamma function as B(p, q) = Γ(p)Γ(q) Γ(p+q) . We note some formulas required in this paper [AS70,Chapter 15]. (3.5) The branch of every z p is the principal value. When b − a ∈ Z, all terms in (3.4) diverge, but an alternative formula is available [AS70,15.3.14]. The formula (3.4), however, is sufficient for our purpose. Similarly, we do not use an alternative formula for (3.5).
The following properties are useful for calculating the Cauchy transforms of beta prime and t-distributions.
(2) Let X be a R-valued random variable. Then, for a = 0 and b ∈ R, (3) If X is a R-valued symmetric random variable, then Proof. Let µ be the distribution of X.
(2) is easy to prove. ( Now we are going to compute the Cauchy transforms of β p,q , β ′ p,q and t q in terms of hypergeometric series. Proof. (1) This is easy from the integral representation (3.1) of the hypergeometric series. ( The formula (3.2) can increase the parameter p + q by 1: This, together with (3.6), leads to the conclusion. (

Free infinite divisibility for beta and beta prime distributions
In order to find a good domain D such that condition (A) holds, the following alternative condition is useful.
(C) There is a connected open set C + ⊂ E ⊂ C such that: (C1) G µ extends to an analytic function in E; The usage of this condition becomes clear in Theorem 4.4, 4.7. Remark 4.6 also explains why this condition is important.
We are going to prove conditions (A) and (C) for beta and beta prime distributions. The following result shows conditions (A1) and (C1), and moreover explicit formulas of the analytic continuation of Cauchy transforms.
. Denoting the analytic continuation by the same symbol G βp,q , we obtain (2) The Cauchy transform G β ′ p,q analytically extends to D bp = E bp := C \ (−∞, 0], and we denote the analytic continuation by the same symbol G β ′ p,q . Then All the powers w → w r are the principal values in the above statements. Proof. (1) Because the density function 1 B(p,q) w p−1 (1 − w) q−1 extends analytically to D b , the Cauchy transform G βp,q also extends to the same region by deforming the contour [0, 1] of the integral to a simple arc γ contained in C − except its endpoints 0, 1. We then consider a simple closed curve γ := γ ∪ [0, 1] with clockwise direction. Take z ∈ C − surrounded by γ, then by residue theorem, we have The left hand side is equal to G βp,q (z) − G βp,q (z). The proof of (2) is similar.
Differential equations for Cauchy transforms are crucial to show (A2) and (C2).
Lemma 4.2. The Cauchy transforms G βp,q , G β ′ p,q satisfy the following differential equations: Proof. Suppose first p, q > 1. Then, by integration by parts, By using the identities Since G βp,q and its derivative depend analytically on p, q > 0, the above differential equation holds for any p, q > 0.
A similar argument is possible for β p,q . Suppose first that p > 1 and then we have The above equation holds for p, q > 0 too because of the analytic dependence on p > 0. The second equality (4.5) follows from the recursive relation The following calculation shows the claim: (2) The Cauchy transform of β ′ p,q satisfies conditions (A2) and (C2) for the domain D bp = E bp for any p, q > 0.
Now we prove the main theorem on beta distributions.
Theorem 4.4. The beta distribution β p,q belongs to class U I in the following cases: (1) Assume moreover that p, q / ∈ Z, p, q / ∈ { 1 2 , 3 2 , 5 2 , · · · } and p + q > 2, because these assumptions simplify the proof. We can recover these exceptions by using the fact that the class U I is closed with respect to the weak convergence [AHb].
In the case (i), we can extend the curve c 1 more because of condition (C2) and the obvious fact G βp,q (u 0 ) = x 0 ; a contradiction to the maximality of c 1 .
Remark 4.6. The computation (4.12) shows that the domain D b satisfies condition (A3) if (p, q) ∈ R := 0, 1 2 ∪ ∞ n=1 2n − 1 2 , 2n + 1 2 2 . Hence, if (p, q) is in the closure of R ∩ {(x, y) : x + y > 2}, the proof is much shorter because one does not need Step 2. However, if (p, q) / ∈ R, the domain D b does not satisfy (A3). In this case, we need condition (C) to find an alternative domain D for condition (A). Such a domain D was realized as D(C).
A similar proof applies for beta prime distributions too.
1. Method based on a local property of probability density function. Given a FID measure µ, the following properties are known thanks to Belinschi and Bercovici: the absolutely continuous part of µ is real analytic wherever it is strictly positive [BB04,Theorem 3.4]; µ has no singular continuous part [ibidem]; µ has at most one atom [BB04,Theorem 3.1].
Now, we will deeply study the real analyticity of the density function of a FID measure. Basic concepts and notations are defined below. Let S (n) (n ∈ Z) denote the open set C \[0, ∞) whose element z is endowed with the argument arg z ∈ (2nπ, 2nπ + 2π). By identifying the slit lim yց0 ([0, ∞) + iy) of S (n) and the slit lim yց0 ([0, ∞) − iy) of S (n−1) for each n, we define a helix-like Riemannian surface S. We express an element z ∈ S uniquely by z = |z|e iθ = re iθ , |z| = r > 0, θ ∈ R. The functions z α = r α e iαθ (α ∈ C) and log z = log r + iθ can be regarded as analytic maps in S. Let S J (R) denote the subset {z ∈ S : arg z ∈ J, 0 < |z| < R} for J ⊂ R, and also S J := ∪ R>0 S J (R). We understand that C + = S (0,π) and (0, ∞) is the half line corresponding to arg z = 0.
We are ready to state the main theorem of this section, which contributes to Theorem 1.1.
Then µ is not FID.
Remark 5.2. A typical function f satisfying the assumptions (ii), (iii) is and absolutely convergent sums of these functions. More restrictively, any real analytic function in a neighborhood of x 0 , vanishing at x 0 , satisfies those assumptions.
Corollary 5.3. The beta distribution β p,q is not FID if p ∈ I or q ∈ I. The beta prime distribution β ′ p,q and the gamma distribution γ p are not FID if p ∈ I.
Example 5.4. (i) If X ∼ β p,q for q ∈ I, then the law of X r is not FID for any r ∈ R \ {0}.
The density function of the law of X r is given by which behaves as c|x − 1| q−1 around x = 1.
(ii) The standard semicircle law w, at x = ±2, corresponds to α = 3 2 which is in the closure of I, but w is FID.
2. Method based on subordination function. We utilize subordination functions introduced by Voiculescu [V93], in order to show the following.
Proposition 5.5. β p,q is not FID for 0 < p, q ≤ 1. 11 Here the assumption α ∈ I is used.
Proof. Let µ be FID and µ t := µ ⊞t . For s ≤ t, a function ω s,t : C + → C + exists so that it satisfies F µs • ω s,t = F µt . The map ω s,t is called the subordination function for (µ t ) t≥0 . We can write ω s,t in terms of F µt : (5.14) It is proved in Theorem 4.6 of [BB05] that ω s,t and hence F µt extends to a continuous function from C + ∪ R into itself. Moreover ω s,t satisfies the inequality Taking the limit s → 0 in (5.14), we get For p, q ∈ (0, 1], the density of β p,q is not continuous at two points 0, 1, so that its reciprocal Cauchy transform is zero at z = 0, 1, which implies that the measure is not FID. 3. Hankel determinants of free cumulants for p = 1 or q = 1. Instead of the analytic method, one can also compute free cumulants (r n ) n≥1 to show that a measure is not FID. The reader is referred to [BG06] and [NS06] for information on free cumulants. The exponential distribution is the limit of D q β 1,q as q → ∞. It is not FID since the 16th Hankel determinant r 2 r 3 r 4 · · · r 17 r 3 r 4 r 5 · · · r 18 · · · · · · · · · r 17 r 18 r 19 · · · r 32 of (r n ) n≥2 is negative. This implies that β 1,q is not FID for large q > 0, because the set of non FID distributions is open with respect to the weak convergence. For smaller q > 0, β 1,q is still not FID; they have negative Hankel determinants for q = 1, 2, · · · , 15.
The conjecture (2) seems to be true from numerical computation. This case however is not covered by Theorem 4.4 because the assumption p + q > 2 is crucial to prove Lemma 4.3.
One may expect that the proof of Theorem 5.1 also applies to any α ∈ ( 1 2 , 3 2 ), but just a slight modification seems not sufficient for that purpose.
6 Free infinite divisibility for Student t-distribution We are going to utilize Proposition 2.9(2) to prove that t-distributions are FID.
Proposition 6.1. The Cauchy transform G tq analytically extends to the domain D st := (C − ∪ H + ) \ i[−1, 0]. We denote the analytic continuation by G tq too. Then Proof. The proof is quite similar to that of Proposition 4.1.
For condition (B2), its proof is based on a recursive differential equation that is quite similar to Lemma 4.2.
By using Lemma 6.2, condition (B2) can be proved as in Lemma 4.3.
Lemma 6.3. The Cauchy transform G tq satisfies condition (B2) for any q > 1 2 in the domain D st .
(2) We are going to show condition (B) for D st . The most important condition is (B3); the others can be shown similarly to the case (1). We show that the limiting values are all in H − ∪ C + ∪ {∞}. (6.6) and (6.7) are computed as in the case (1) and they belong to H − ∪ C + ∪ {∞}.
Conjecture 6.6. The t-distribution t q is FID (and more strongly in class U I) for any q > 1 2 .

Free divisibility indicator of symmetric distribution
A family of maps {B t } t≥0 is defined on the set of Borel probability measures P [BN08]: where ⊎ is Boolean convolution [SW97] and the probability measure µ ⊎t (t ≥ 0) is defined by F µ ⊎t (z) = (1 − t)z + tF µ (z). These maps become a flow: A probability measure µ is FID if and only if φ(µ) ≥ 1 [BN08]. The following property is known [AHc]: Hence, when φ(µ) < ∞, µ ⊎t is FID for small t > 0, and the free divisibility indicator measures the time when the Boolean time evolution breaks the free infinite divisibility. We will give a method for calculating the quantity φ(µ).
The free divisibility indicator is not continuous with respect to the weak convergence, as one can observe from Wigner's semicircle law w t with mean 0 and variance t. Indeed, φ(w t ) = 1 for any t > 0, while φ(w 0 ) = ∞ (see [BN08] for this computation). Hence, Proposition 7.2 is not sufficient to calculate the exact value of the free divisibility indicator of Gaussian which is the weak limit of scaled ultraspherical distributions or t-distributions. Here we will show that the value is exactly 1 for the Gaussian distribution. The classical infinite divisibility of the Boolean power of Gaussian is also studied here.
Proposition 7.4. Let g be the standard Gaussian.
Proof. (1) Some properties shown below are known in [BBLS11], but we try to make this proof self-contained. We are going to check Lemma 7.1(1). Let f (y) denote the function 1 i F g (iy). The function F g extends to iR analytically and does not have a pole in iR since F g (z) = lim p→∞ F D √ 2q (tq) (z) locally uniformly in iR. This convergence holds not only in i(0, ∞) but also in i(−∞, 0] by changing the contour R of the integral in G D √ 2q (tq) to an arc in C − ∪ R as in Proposition 4.1. Because 1 i F D √ 2q (tq) (iy) > 0 for y ∈ (− √ 2q, ∞) and it is strictly increasing as proved in Lemma 6.4, one also has f (y) > 0 for y ∈ R. The function f satisfies the differential equation f ′ (y) = f (y) 2 − yf (y) (7.4) as proved in [BBLS11,Eq. (3.6)], which also follows from a limit of Lemma 6.2(2). If y > 0, then f (y) > y from the basic property of a reciprocal Cauchy transform, and hence f ′ (y) = f (y)(f (y) − y) > 0. If y ≤ 0, then f ′ (y) > 0 from the fact f (y) > 0 and (7.4). Hence, f ′ (y) > 0 for every y ∈ R.
(2) By shifting the contour by −i , one can write We divide the integral into two parts. First we find y 2 e − 1 2 y 2 + 1 2 dy → 0 as x → ∞.
Next, we have sup y∈(−∞, from the dominated convergence theorem. By symmetry, we conclude xG g (x) → 1 as |x| → ∞. From the Stieltjes inversion formula, the density of g ⊎t can be written as For each t > 0, the above density behaves like t √ 2π e − x 2 2 for large |x| > 0 since xG g (x) → 1. Any classically infinitely divisible distribution with Gaussian-like tail behavior must be exactly a Gaussian (see Corollary 9.9 of [HS04]). Therefore, g ⊎t is not infinitely divisible for t = 0, 1.
Finally we show a general result which in particular enables us to compute the free divisibility indicator of the distribution w ⊞ g. This measure appeared as the spectral distribution of large random Markov matrices [BDJ06].