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May 2011 A refined factorization of the exponential law
P. Patie
Bernoulli 17(2): 814-826 (May 2011). DOI: 10.3150/10-BEJ292


Let $ξ$ be a (possibly killed) subordinator with Laplace exponent $ϕ$ and denote by $I_ϕ = ∫_0^∞e^{−ξ_s} ds$, the so-called exponential functional. Consider the positive random variable $I_{ψ1}$ whose law, according to Bertoin and Yor [Electron. Comm. Probab. 6 (2001) 95–106], is determined by its negative entire moments as follows: $$\mathbb {E}[I_{\psi_{1}}^{-n}]=\prod_{k=1}^{n}\phi(k),\qquad n=1,2,\ldots.$$ In this note, we show that $I_{ψ1}$ is a positive self-decomposable random variable whenever the Lévy measure of $ξ$ is absolutely continuous with a monotone decreasing density. In fact, $I_{ψ1}$ 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 e: $$I_{\phi}/I_{\psi_{1}}\stackrel {\mathrm {(d)}}{=}{\mathbf {e}},$$ where $I_{ψ1}$ is taken to be independent of $I_ϕ$. 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 $S(α)^α$ is a self-decomposable random variable, where $S(α)$ is a positive stable random variable of index $α ∈ (0, 1)$.


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P. Patie. "A refined factorization of the exponential law." Bernoulli 17 (2) 814 - 826, May 2011.


Published: May 2011
First available in Project Euclid: 5 April 2011

zbMATH: 1253.60020
MathSciNet: MR2787616
Digital Object Identifier: 10.3150/10-BEJ292

Rights: Copyright © 2011 Bernoulli Society for Mathematical Statistics and Probability


Vol.17 • No. 2 • May 2011
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