Random walks are determined by their trace on the positive half-line
Annales Henri Lebesgue, Volume 3 (2020) , pp. 1389-1397.

KeywordsRandom walk, Lévy process, Wiener–Hopf factorisation, Nevanlinna class

### Abstract

We prove that the law of a random walk ${X}_{n}$ is determined by the one-dimensional distributions of $max\left({X}_{n},0\right)$ for $n=1,2,...\phantom{\rule{0.166667em}{0ex}}$, 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.

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