A refined factorization of the exponential law

Abstract

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 $α ∈ (0, 1)$.