On classical and free stable laws

TL;DR

This paper derives explicit representations for the densities of classical and free stable laws using advanced special functions, revealing connections to Beta distributions and asymptotic behaviors.

Contribution

It provides a new explicit Fox-H function representation of the density of a transformed classical stable law and clarifies its relation to free probability densities.

Findings

Representation of the density as a Fox-H function.

Connection of special cases to Meijer G-functions and Beta distributions.

Asymptotic behavior of Bernstein measures as alpha approaches zero.

Abstract

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 degenerate thereby agrees with Cressie's result for the asymptotic behaviour of stable distributions for small values of $\alpha$. Coming to free probability, our result makes more explicit that of Biane on the density of its free analog. The paper is closed with analytic arguments explaining the occurence of the Kanter's random variable in both the classical and the free settings.