Random walks are determined by their trace on the positive half-line

Kwaśnicki, Mateusz

Annales Henri Lebesgue, Volume 3 (2020), pp. 1389-1397.

We prove that the law of a random walk $X_n$ is determined by the one-dimensional distributions of $max(X_n,0)$ for $n=1,2,...$, as conjectured recently by Loïc Chaumont and Ron Doney. Equivalently, the law of $X_n$ is determined by its upward space-time Wiener–Hopf factor. Our methods are complex-analytic.

Nous démontrons que la loi d’une marche aléatoire $X_n$ est déterminée par les distributions de $max(X_n,0)$ pour $n=1,2,...$, comme l’avaient conjecturé récemment Loïc Chaumont et Ron Doney. De manière équivalente, la loi de $X_n$ est déterminée par son facteur de Wiener–Hopf espace-temps ascendant. Nos méthodes relèvent de l’analyse complexe.

Classification: 60G50, 60G51, 45E10, 30H15
Keywords: Random walk, Lévy process, Wiener–Hopf factorisation, Nevanlinna class

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