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

中文翻译：

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关于由它们的向上时空维纳霍普夫因子确定的分布

根据 Wiener-Hopf 分解，在 $$\mathbb {R}$$ R 上的任何概率分布 $$\mu $$ μ 的特征函数 $$\varphi $$ φ 可以以独特的方式分解为 $$ \begin{aligned} 1-s\varphi (t)=[1-\chi _-(s,it)][1-\chi _+(s,it)],\quad |s|\le 1, \,t\in \mathbb {R}\,, \end{aligned}$$ 1 - s φ ( t ) = [ 1 - χ - ( s , it ) ] [ 1 - χ + ( s , it ) ] , | | ≤ 1 , t ∈ R , 其中 $$\chi _-(e^{iu},it)$$ χ - ( e iu , it ) 和 $$\chi _+(e^{iu},it)$ $ χ + ( e iu , it ) 是 $$\mathbb {Z}_+\times (-\infty ,0)$$ Z + × ( - ∞ , 0 ) 和 $$ 中可能有缺陷的分布的特征函数\mathbb {Z}_+\times [0,\infty )$$ Z + × [ 0 , ∞ ) 分别。我们证明了 $$\mu $$ μ 可以通过向上因子 $$\chi _+(s,it)$$ χ + ( s , it ) , $$s\in [0, 1)$$s ∈ [ 0 , 1 ) , $$t\in \mathbb {R}$$ t ∈ R 在许多情况下包括以下情况： 1. $$\mu $$ μ 有一些指数矩；2. 函数 $$t\mapsto \mu (t,\infty )$$ t ↦ μ ( t , ∞ ) 在 $$(0,\infty )$$ ( 0 , ∞ ) 上是完全单调的；3. $$[0,\infty )$$ [ 0 , ∞ ) 上 $$\mu $$ μ 的密度承认 $$\mathbb {R}$$ R 上的解析延拓。我们推测任何概率分布实际上都以其向上的因素为特征。该猜想等价于以下内容： 在 $$\mathbb {R}$$ R 上的任何概率测度 $$\mu $$ μ 其支持度不包含在 $$(-\,\infty ,0)$$ ( - ∞ , 0 ) 由其卷积幂决定 $$\mu ^{*n}$$ μ ∗ n , $$n\ge 1$$ n ≥ 1 限制为 $$[0,\infty )$$ [ 0 , ∞ ) 。我们表明，在许多情况下，$$\mu $$ μ 和 $$\mu ^{*2}$$ μ ∗ 2 的唯一知识仅限于 $$[0,\infty )$$ [ 0 , ∞ ) 实际上足以确定 $$\mu $$ μ 。然后我们在无限可分分布的框架中研究类似的问题。