A refined factorization of the exponential law

Abstract

Let $\xi$ be a (possibly killed) subordinator with Laplace exponent $\varphi$ and denote by $I_{\varphi }={\int }_0^{\infty }{\mathrm{e}}^{-{\xi }_s}\mathrm{d}s$, the so-called exponential functional. Consider the positive random variable $I_{{\psi }_1}$ whose law, according to Bertoin and Yor [Electron. Comm. Probab. 6 (2001) 95--106], is determined by its negative entire moments as follows: \[\mathbb {E}[I_{ψ_1}^{-n}]=\prod_{k=1}^nϕ(k),\qquad n=1,2,...\] In this note, we show that $I_{{\psi }_1}$ is a positive self-decomposable random variable whenever the Lévy measure of $\xi$ is absolutely continuous with a monotone decreasing density. In fact, $I_{{\psi }_1}$ is identified as the exponential functional of a spectrally negative (sn, for short) Lévy process. We deduce from Bertoin and Yor [Electron. Comm. Probab. 6 (2001) 95--106] the following factorization of the exponential law $\mathbf{e}$: \[I_ϕ/I_{ψ_1}\stackrel{\mathrm {(d)}}{=}{\mathbf {e}},\] where $I_{{\psi }_1}$ is taken to be independent of $I_{\varphi }$. We proceed by showing an identity in distribution between the entrance law of an sn self-similar positive Feller process and the reciprocal of the exponential functional of sn Lévy processes. As a by-product, we obtain some new examples of the law of the exponential functionals, a new factorization of the exponential law and some interesting distributional properties of some random variables. For instance, we obtain that $S(\alpha {)}^{\alpha }$ is a self-decomposable random variable, where $S(\alpha )$ is a positive stable random variable of index $\alpha \in (0,1)$.

Published in at http://dx.doi.org/10.3150/10-BEJ292 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)

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• Weak convergence of positive self-similar Markov processes and overshoots of Lévy processes
• A transformation for Lévy processes with one-sided jumps and applications