On distributions determined by their upward, space-time Wiener-Hopf factor

Abstract

According to the Wiener-Hopf factorization, the characteristic function $\varphi$ of any probability distribution $\mu$ on $\mathbb{R}$ can be decomposed in a unique way as \[1-s\varphi(t)=[1-\chi_-(s,it)][1-\chi_+(s,it)]\,,\;\;\;|s|\le1,\,t\in\mathbb{R}\,,\] where $\chi_-(e^{iu},it)$ and $\chi_+(e^{iu},it)$ are the characteristic functions of possibly defective distributions in $\mathbb{Z}_+\times(-\infty,0)$ and $\mathbb{Z}_+\times[0,\infty)$, respectively. We prove that $\mu$ can be characterized by the sole data of the upward factor $\chi_+(s,it)$, $s\in[0,1)$, $t\in\mathbb{R}$ in many cases including the cases where: 1) $\mu$ has some exponential moments; 2) the function $t\mapsto\mu(t,\infty)$ is completely monotone on $(0,\infty)$; 3) the density of $\mu$ on $[0,\infty)$ admits an analytic continuation on $\mathbb{R}$. We conjecture that any probability distribution is actually characterized by its upward factor. This conjecture is equivalent to the following: {\it Any probability measure $\mu$ on $\mathbb{R}$ whose support is not included in $(-\infty,0)$ is determined by its convolution powers $\mu^{*n}$, $n\ge1$ restricted to $[0,\infty)$}. We show that in many instances, the sole knowledge of $\mu$ and $\mu^{*2}$ restricted to $[0,\infty)$ is actually sufficient to determine $\mu$. Then we investigate the analogous problem in the framework of infinitely divisible distributions.