On distributions determined by their upward, space-time Wiener-Hopf factor
Résumé
According to the Wiener-Hopf factorization, the characteristic function ϕ of any probability distribution µ on R can be decomposed in a unique way as 1 − sϕ(t) = [1 − χ − (s, it)][1 − χ + (s, it)] , |s| ≤ 1, t ∈ R , where χ − (e iu , it) and χ + (e iu , it) are the characteristic functions of possibly defective distributions in Z + × (−∞, 0) and Z + × [0, ∞), respectively. We prove that µ can be characterized by the sole data of the upward factor χ + (s, it), s ∈ [0, 1), t ∈ R in many cases including the cases where: 1) µ has some exponential moments; 2) the function t → µ(t, ∞) is completely monotone on (0, ∞); 3) the density of µ on [0, ∞) admits an analytic continuation on R. We conjecture that any probability distribution is actually characterized by its upward factor. This conjecture is equivalent to the following: Any probability measure µ on R whose support is not included in (−∞, 0) is determined by its convolution powers µ * n , n ≥ 1 restricted to [0, ∞). We show that in many instances, the sole knowledge of µ and µ * 2 restricted to [0, ∞) is actually sufficient to determine µ. Then we investigate the analogous problem in the framework of infinitely divisible distributions.
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