Free Infinite Divisibility of Free Multiplicative Mixtures of the Wigner Distribution

Abstract

Let I * and I ⊞ be the classes of all classical infinitely divisible distributions and free infinitely divisible distributions, respectively, and let Λ be the Bercovici-Pata bijection between I * and I ⊞ . The class type W of symmetric distributions in I ⊞ that can be represented as free multiplicative convolutions of the Wigner distribution is studied. A characterization of this class under the condition that the mixing distribution is 2-divisible with respect to free multiplicative convolution is given. A correspondence between symmetric distributions in I ⊞ and the free counterpart under Λ of the positive distributions in I * is established. It is shown that the class type W does not include all symmetric distributions in I ⊞ and that it does not coincide with the image under Λ of the mixtures of the Gaussian distribution in I * . Similar results for free multiplicative convolutions with the symmetric arcsine measure are obtained. Several well-known and new concrete examples are presented.

On the non-classical infinite divisibility of power semicircle distributions

“…This conjecture has recently been proved to be true in [4]. Moreover, we conjecture that the classical Gaussian distribution is a free multiplicative convolution of the semicircle distribution in the sense of [3], [40]. Remark 4.6.…”

On Free Regular and Bondesson Convolution Semigroups

“…The following corollary of Theorem 2 shows that the cumulative distribution function of any measure in the Bondesson class BO ♭ can be expressed in terms of the corresponding ⊞-convolution semigroup. This result is easily derived by rescaling t → t/w and taking derivative d/dw of both sides of identity (12). Corollary 1.…”

“…The class of free regular measures will be denoted by I ⊞ r+ . Free regular distributions were introduced in [12] and studied later in [1,13]. Note that free regular measures are defined differently in [1], but our definition 3 is an equivalent one (see [1][Theorem 4.2]).…”

Regular variation and free regular infinitely divisible laws

“…It is a pertinent question that whether the conclusion involving Marchenko-Pastur law can be replaced by the standard Wigner's semicircle law, w. The measures of the form w ⊠ ρ for some ρ ∈ M + has appeared as the limiting spectral distributions of random matrices (see Anderson and Zeitouni [2008], Chakrabarty et al [2016Chakrabarty et al [ , 2018b), free type W distributions (see Pérez-Abreu and Sakuma [2012]) and in several other places.…”

“…Assume (1). Theorem 22 of Pérez-Abreu and Sakuma [2012] says that for ρ ∈ M + and w be the standard Wigner measure, then ρ 0 = ρ ⊠ ρ is a free regular infinitely divisible probability measure if and only if µ = w ⊠ ρ is a symmetric free infinitely divisible probability measure. Now from Lemma 8 of Arizmendi E. and Pérez-Abreu [2009] we get…”