Volume 23, 2019
|Page(s)||82 - 111|
|Published online||14 March 2019|
On the law of homogeneous stable functionals
Centre International de Valbonne,
190 Rue Frédéric Mistral,
2 Laboratoire Paul Painlevé, Université de Lille, Cité Scientifique, 59655 Villeneuve d’Ascq, France.
* Corresponding author: firstname.lastname@example.org
Accepted: 20 December 2018
Let A be the Lq-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  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.
Mathematics Subject Classification: 60G51 / 60G52
Key words: Beta random variable / exponential functional / homogeneous functional / infinite divisibility / stable Lévy process / time-change
© EDP Sciences, SMAI 2019
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