指数律的精细分解
A refined factorization of the exponential law
摘要：
设$ξ$是一个（可能是killed）具有拉普拉斯指数$ϕ$的从属子，用$I_ϕ=\int_0^{\infty}\mathrm{e}^{-ξ_s}\,\mathrm{d}s$表示，即所谓的指数泛函。考虑正随机变量$I_{ψ_1}$，根据Bertoin和Yor [Electron. Comm. Probab. 6（2001）95--106]，由它的负整矩确定如下：\[\mathbb {E}[I_{ψ_1}^{-n}]=\prod_{k=1}^nϕ(k),\qquad n=1,2,...\]本文证明了：只要$ξ$的Lévy测度是绝对连续且密度单调递减的，则$I_{ψ_1}$是正的自分解随机变量.事实上，$I_{ψ_1}$被确定为一个谱负（简称sn）Lévy过程的指数泛函。我们推断从Bertoin和Yor [电子。6（2001）95--106]，指数律${\mathbf {e}}$的下列因式分解：\[I_ϕ/I_{ψ_1}\stackrel{\mathrm {(d)}}{=}{\mathbf {e}},\]其中，$I_{ψ_1}$被认为独立于$I_ϕ$。我们继续通过显示一个身份之间的分布的入口法的一个sn自相似的正费勒过程和倒数的指数功能的sn Lévy过程。作为副产品，我们得到了一些新的例子的法律的指数泛函，一个新的因式分解的指数律和一些有趣的分布性质的随机变量。例如，我们得到了$S(α)^α$是一个自分解的随机变量，其中$S(α)$是一个正稳定的随机变量的指标$α\in(0,1)$。
英文摘要：
Let $ξ$ be a (possibly killed) subordinator with Laplace exponent $ϕ$ and denote by $I_ϕ=\int_0^{\infty}\mathrm{e}^{-ξ_s}\,\mathrm{d}s$, the so-called exponential functional. Consider the positive random variable $I_{ψ_1}$ 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 $I_{ψ_1}$ is a positive self-decomposable random variable whenever the Lévy measure of $ξ$ is absolutely continuous with a monotone decreasing density. In fact, $I_{ψ_1}$ 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 $I_{ψ_1}$ 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 $α\in(0,1)$.
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