
TL;DR
This paper refines the understanding of exponential functionals of subordinators, showing they are self-decomposable under certain conditions, and introduces new factorizations and distributional properties related to exponential laws and stable variables.
Contribution
It identifies conditions under which exponential functionals are self-decomposable and provides a new factorization of the exponential law involving these functionals.
Findings
$I_{ ext{psi}_1}$ is self-decomposable when the Lévy measure has a decreasing density.
A new distributional identity links entrance laws of sn self-similar processes to exponential functionals.
The paper introduces a novel factorization of the exponential law and properties of stable distributions.
Abstract
Let be a (possibly killed) subordinator with Laplace exponent and denote by , the so-called exponential functional. Consider the positive random variable whose law, according to Bertoin and Yor [Electron. Comm. Probab. 6 (2001) 95--106], is determined by its negative entire moments as follows: \[\mathbb {E}[I_{\psi_1}^{-n}]=\prod_{k=1}^n\phi(k),\qquad n=1,2,...\] In this note, we show that is a positive self-decomposable random variable whenever the L\'{e}vy measure of is absolutely continuous with a monotone decreasing density. In fact, is identified as the exponential functional of a spectrally negative (sn, for short) L\'{e}vy process. We deduce from Bertoin and Yor [Electron. Comm. Probab. 6 (2001) 95--106] the following factorization of the exponential law ${\mathbf…
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