On the law of homogeneous stable functionals

¹ Centre International de Valbonne, 190 Rue Frédéric Mistral, 06560 Valbonne, France.
² Laboratoire Paul Painlevé, Université de Lille, Cité Scientifique, 59655 Villeneuve d’Ascq, France.

^* Corresponding author: simon@math.univ-lille1.fr

Received: 16 January 2017
Accepted: 20 December 2018

Abstract

Let A be the L^q-functional of a stable Lévy process starting from one and killed when crossing zero. We observe that 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 transformation, and an explicit computation performed in [38] on perpetuities of hypergeometric Lévy processes. This representation allows us to retrieve several factorizations previously shown by various authors, and also to derive new ones. We emphasize the connections between 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 A related to infinite divisibility, self-decomposability, and the generalized Gamma convolution.