On the law of homogeneous stable functionals

TL;DR

This paper provides a novel representation of a stable Le9vy process functional as a quotient of infinite products of Beta variables, revealing new factorizations and distributional properties.

Contribution

It introduces a new representation of the homogeneous stable functional using Beta random variables and explores its connections and properties.

Findings

Representation as quotient of Beta products

Retrieval of known factorizations and new ones

Analysis of distributional properties like infinite divisibility

Abstract

Let ${\mathcal A}$ be the ${\mathcal L}^q-$functional of a stable L\'evy process starting from one and killed when crossing zero. We observe that ${\mathcal A}$ can be represented as the independent quotient of two infinite products of renormalized Beta random variables. The proof relies on Markovian time change, the Lamperti transform, and an explicit computation on perpetuities of hypergeometric L\'evy processes previously obtained by Kuznetsov and Pardo. This representation allows to retrieve several factorizations previously obtained by various authors, and also to derive new ones. We emphasize the connections between ${\mathcal A}$ and more standard positive random variables. We also investigate the law of Riemannian integrals of stable subordinators. Finally, we derive several distributional properties of ${\mathcal A}$ related to infinite divisibility, self-decomposability, and the generalized Gamma convolutions.