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 $$\begin{aligned} 1-s\varphi (t)=[1-\chi _-(s,it)][1-\chi _+(s,it)],\quad |s|\le 1,\,t\in \mathbb {R}\,, \end{aligned}$$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: We conjecture that any probability distribution is actually characterized by its upward factor. This conjecture is equivalent to the following: Any probability measure$$\mu $$on$$\mathbb {R}$$whose support is not included in$$(-\,\infty ,0)$$is determined by its convolution powers$$\mu ^{*n}$$, $$n\ge 1$$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.