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On distributions determined by their upward, space-time Wiener-Hopf factor. (English) Zbl 1455.60031
Summary: 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:
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: 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.

##### MSC:
 60E05 Probability distributions: general theory 60A10 Probabilistic measure theory 60E10 Characteristic functions; other transforms
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##### References:
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