Related papers: The normal distribution is $\boxplus$-infinitely d…
The extension $k \mapsto \mu^{\boxplus k}$ of the concept of a free convolution power to the case of non-integer $k \geq 1$ was introduced by Bercovici-Voiculescu and Nica-Speicher, and related to the minor process in random matrix theory.…
We study the class $\mathcal{M}_{\mathrm{ratio}}$ of those probability distributions for which the free $R$-transforms are rational functions. This class is closed under the additive free convolution, additive free powers and under the…
We construct a random matrix model for the bijection \Psi between clas- sical and free infinitely divisible distributions: for every d\geq1, we associate in a quite natural way to each *-infinitely divisible distribution \mu a distribution…
For a variety of pattern-avoiding classes, we describe the limiting distribution for the number of fixed points for involutions chosen uniformly at random from that class. In particular we consider monotone patterns of arbitrary length as…
We consider a new class $\boldsymbol{Q}$ of distribution functions $F$ that have the property of rational-infinite divisibility: there exist some infinitely divisible distribution functions $F_1$ and $F_2$ such that $F_1=F*F_2$. A…
We introduce a finite version of free probability and show the link between recent results using polynomial convolutions and the traditional theory of free probability. One tool for accomplishing this is a seemingly new transformation that…
Probability distributions supported on the simplex enjoy a wide range of applications across statistics and machine learning. Recently, a novel family of such distributions has been discovered: the continuous categorical. This family enjoys…
Asymptotic properties of finitely generated subgroups of free groups, and of finite group presentations, can be considered in several fashions, depending on the way these objects are represented and on the distribution assumed on these…
We provide necessary and sufficient conditions on the characteristics of an infinitely divisible distribution under which its characteristic function $\phi$ decays polynomially. Under a mild regularity condition this polynomial decay is…
We investigate certain analytical properties of the free $\alpha-$stable densities on the line. We prove that they are all classically infinitely divisible when $\alpha\le 1$, and that they belong to the extended Thorin class when $\alpha…
We propose an extension of the classical variational theory of evolution equations that accounts for dynamics also in possibly non-reflexive and non-separable spaces. The pivoting point is to establish a novel variational structure, based…
For the class of free-infinitely divisible transforms are introduced three families of increasing Urbanik type subclasses of those transforms. They begin with the class of free-normal transforms and end up with the whole class of…
This paper introduces the class of selfdecomposable distributions concerning Boolean convolution. A general regularity property of Boolean selfdecomposable distributions is established; in particular the number of atoms is at most two and…
A Wright function based framework is proposed to combine and extend several distribution families. The $\alpha$-stable distribution is generalized by adding the degree of freedom parameter. The PDF of this two-sided super distribution…
Consider the Kontsevich $\star$-product on the symmetric algebra of a finite dimensional Lie algebra $\mathfrak g$, regarded as the algebra of distributions with support 0 on $\mathfrak g$. In this paper, we extend this $\star$-product to…
We study a new class of so-called quasi-infinitely divisible laws, which is a wide natural extension of the well known class of infinitely divisible laws through the L\'evy--Khinchine type representations. We are interested in criteria of…
The classical infinite divisibility of distributions related to eigenvalues of some random matrix ensembles is investigated. It is proved that the $\beta$-Tracy-Widom distribution, which is the limiting distribution of the largest…
In this article we study the influence of regularly varying probability measures on additive and multiplicative Boolean convolutions. We introduce the notion of Boolean subexponentiality (for additive Boolean convolution), which extends the…
The infinitely-many-neutral-alleles model has recently been extended to a class of diffusion processes associated with Gibbs partitions of two-parameter Poisson-Dirichlet type. This paper introduces a family of infinite-dimensional…
A probability distribution $\mu$ on $\mathbb{R}^d$ is quasi-infinitely divisible if its characteristic function has the representation $\widehat{\mu} = \widehat{\mu_1}/\widehat{\mu_2}$ with infinitely divisible distributions $\mu_1$ and…