Stable laws and domains of attraction in free probability theory

We investigate the analytical properties of free stable distributions and discover many connections with the classical stable distributions. Our main results include an explicit formula for the Mellin transform, series representations for the characteristic function and for the density of the free stable distribution; all of these explicit formulas exhibit close resemblance to the corresponding expressions for classical stable distributions. As applications of these results w...

Based on a new analytical approach to the definition of additive free convolution on probability measures on the real line we prove free analogs of limit theorems for sums for non-identically distributed random variables in classical Probability Theory.

In this paper additive bi-free convolution is defined for general Borel probability measures, and the limiting distributions for sums of bi-free pairs of selfadjoint commuting random variables in an infinitesimal triangular array are determined. These distributions are characterized by their bi-freely infinite divisibility, and moreover, a transfer principle is established for limit theorems in classical probability theory and Voiculescu's bi-free probability theory. Complete...

The phenomenon of superconvergence is proved for all freely infinitely divisible distributions. Precisely, suppose that the partial sums of a sequence of free identically distributed, infinitesimal random variables converge in distribution to a nondegenerate freely infinitely divisible law. Then the distribution of the sum becomes Lebesgue absolutely continuous with a continuous density in finite time, and this density can be approximated by that of the limit law uniformly, a...

We introduce a large and flexible class of discrete tempered stable distributions, and analyze the domains of attraction for both this class and the related class of positive tempered stable distributions. Our results suggest that these are natural models for sums of independent and identically distributed random variables with tempered heavy tails tails, i.e. tails that appear to be heavy up to a point, but ultimately decay faster.

This paper describes the quality of convergence to an infinitely divisible law relative to free multiplicative convolution. We show that convergence in distribution for products of identically distributed and infinitesimal free random variables implies superconvergence of their probability densities to the density of the limit law. Superconvergence to the marginal law of free multiplicative Brownian motion at a specified time is also studied. In the unitary case, the supercon...

This paper derives sufficient conditions for superconvergence of sums of bounded free random variables and provides an estimate for the rate of superconvergence.

We consider different limit theorems for additive and multiplicative free L\'evy processes. The main results are concerned with positive and unitary multiplicative free L\'evy processes at small time, showing convergence to log free stable laws for many examples. The additive case is much easier, and we establish the convergence at small or large time to free stable laws. During the investigation we found out that a log free stable law with index $1$ coincides with the Dykema...

We continue here the study of free extreme values begun in Ben Arous and Voiculescu (2006). We study the convergence of the free point processes associated with free extreme values to a free Poisson random measure (Voiculescu (1998), Barndorff-Nielsen and Thorbjornsen (2005)). We relate this convergence to the free extremal laws introduced in Ben Arous and Voiculescu (2006) and give the limit laws for free order statistics.

We derive the representative Bernstein measure of the density of $(X_{\alpha})^{-\alpha/(1-\alpha)}, 0 < \alpha < 1$, where $X_{\alpha}$ is a positive stable random variable, as a Fox-H function. When $1-\alpha = 1/j$ for some integer $j \geq 2$, the Fox H-function reduces to a Meijer G-function so that the Kanter's random variable (see below) is closely related to a product of $(j-1)$ independent Beta random variables. When $\alpha$ tends to 0, the Bernstein measure becomes ...