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

## TL;DR

This paper proves that the distribution of a random walk is uniquely determined by the distributions of its positive parts, confirming a recent conjecture and utilizing complex-analytic methods.

## Contribution

It establishes that the law of a random walk is uniquely determined by its positive half-line distributions, resolving a recent conjecture.

## Key findings

- Law of a random walk determined by positive parts
- Confirmation of recent conjecture by Chaumont and Doney
- Use of complex-analytic methods

## Abstract

We prove that the law of a random walk $X_n$ is determined by the one-dimensional distributions of $\max(X_n, 0)$ for $n = 1, 2, \ldots$, as conjectured recently by Lo\"ic 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.