On Distributions Determined by Their Upward, Space–Time Wiener–Hopf Factor

Loïc Chaumont () and Ron Doney ()
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Loïc Chaumont: Université d’Angers
Ron Doney: University of Manchester

Journal of Theoretical Probability, 2020, vol. 33, issue 2, 1011-1033

Abstract: Abstract 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-sφ(t)=[1-χ-(s,it)][1-χ+(s,it)],|s|≤1,t∈R,where $$\chi _-(e^{iu},it)$$χ-(eiu,it) and $$\chi _+(e^{iu},it)$$χ+(eiu,it) 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,it), $$s\in [0,1)$$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$$\mu $$μon$$\mathbb {R}$$Rwhose support is not included in$$(-\,\infty ,0)$$(-∞,0)is determined by its convolution powers$$\mu ^{*n}$$μ∗n, $$n\ge 1$$n≥1restricted 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.

Keywords: Wiener–Hopf factors; Convolution powers; Exponential moments; Completely monotone function; 60A10; 60E05 (search for similar items in EconPapers)
Date: 2020
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DOI: 10.1007/s10959-019-00889-x

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