Some properties of the free stable distributions

Abstract

We investigate certain analytical properties of the free α−stable densities on the line. We prove that they are all classically infinitely divisible when α ≤ 1, and that they belong to the extended Thorin class when α ≤ 3/4. The Lévy measure is explicitly computed for α = 1, showing that the free 1-stable random variables are not Thorin except in the drifted Cauchy case. In the symmetric case we show that the free stable densities are not infinitely divisible when α > 1. In the one-sided case we prove, refining unimodality, that the densities are whale-shaped that is their successive derivatives vanish exactly once. Finally, we derive a collection of results connected to the fine structure of the one-sided free stable densities, including a detailed analysis of the Kanter random variable, complete asymptotic expansions at zero, a new identity for the Beta-Gamma algebra, and several intrinsic properties of whale-shaped densities.where X 1 , X 2 are free independent copies of a random variable X with density f , a, b are arbitrary positive real numbers, c is a positive real number depending on a, b, and d is a real number. As in the classical framework, it turns out that there exist solutions to (1) only if c = (a α + b α ) 1/α for some fixed α ∈ (0, 2]. We will be mostly concerned with free strictly stable densities, which correspond to the case d = 0. In this framework the Voiculescu transform of f writes, up to multiplicative normalization,We refer e.g. to [43] for some background on the free additive convolution, to [11] for the original solution to the equation (1), and to the introduction of [29] for the above parametrization (α, ρ), which mimics that of the strict classical framework. Let us also recall that free stable laws appear as limit distributions of spectra of large random matrices with possibly unbounded variance -see [10,18], and that their domains of attraction have been fully characterized in [12,13]. In the following, we will denote by X α,ρ the random variable whose Voiculescu transform is given by (2), and set f α,ρ for its density. The analogy with the classical case extends to the fact, observed in Corollary 1.3 of [29], that with our parametrization one hasThe explicit form of the Voiculescu transform also shows that X α,ρ d = −X α,1−ρ . In this paper, some focus will put on the one-sided case and we will use the shorter notations X α,1 = X α and f α,1 = f α .Throughout, the random variable X α,ρ will be mostly handled as a classical random variable via its 2010 Mathematics Subject Classification. 60E07; 46L54; 62E15; 33E20.

Homomorphisms relative to additive convolutions and max-convolutions: Free, boolean and classical cases

“…so that Ξ(b α ) is the Dagum distribution, which is a boolean extreme value distribution [34,Corollary 4.1]. It coincides with (e −1 ⊛ e) 1/α as observed in [20,Remark 6], which also appears in Section 4.2; see (4.8).…”

An Operad of Non-commutative Independences Defined by Trees

“…As mentioned in the introduction, we did not fully develop the complex-analytic viewpoint on T -free independences. Moreover, at least in the scalar-valued setting, the complex-analytic viewpoint should allow the study of T -free convolution of arbitrary probability measures, discovery of the optimal rate of convergence in the central limit theorem (see Remark 8.14), and the classification of infinitely divisible and stable distributions with unbounded support (as in [20], [13], [8], [32]). We would also like to know under what conditions the estimate max j (rad(µ j )) ≤ rad(⊞ T (µ 1 , .…”

A new class of bell-shaped functions

“…It combines an application of Post's formula for the inverse Laplace transform of the Cauchy-Stieltjes transform of f with some ideas developed by Hirschman in [3]. Our proof of Theorem 1.3 has its roots in Proposition 10 in [2], which asserts that whale-shaped functions have the representation similar to that in Theorem 1.1, with ϕ taking values in [0, 2].…”