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

Abstract

According to the Wiener-Hopf factorization, the characteristic function $\phi$ of any probability distribution $\mu$ on $\mathbb{R}$ can be decomposed in a unique way as \[1-sφ(t)=[1-χ_-(s,it)][1-χ_+(s,it)]\,,\;\;\;|s|\le1,\,t\in\mathbb{R}\,,\] 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 ${\mu }^{\ast n}$, $n\ge 1$ restricted to $[0,\infty )$}. We show that in many instances, the sole knowledge of $\mu$ and ${\mu }^{\ast 2}$ restricted to $[0,\infty )$ is actually sufficient to determine $\mu$. Then we investigate the analogous problem in the framework of infinitely divisible distributions.