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

*(English)*Zbl 1455.60031Summary: 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.

\[\begin{aligned}1-s\varphi(t)=[1-\chi_-(s,it)][1-\chi_+(s,it)],\quad|s|\le 1,\,t\in\mathbb{R}\, \end{aligned}\]

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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\textit{L. Chaumont} and \textit{R. Doney}, J. Theor. Probab. 33, No. 2, 1011--1033 (2020; Zbl 1455.60031)

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