# Exact Coupling of Random Walks on Polish Groups

## TL;DR

This paper characterizes when exact coupling of random walks on Polish groups is possible, providing necessary and sufficient conditions in the Abelian case, and analyzes the convergence rates and structure of such couplings.

## Contribution

It offers a complete characterization of successful exact coupling conditions for random walks on Abelian Polish groups, including convergence rates and measurable structure.

## Key findings

- Successful exact coupling characterized by $orall n, ext{support}(
u^n)
i x$
- Total variation distance decays as $O(n^{-1/2})$ or exponentially
- Solved a problem posed by H. Thorisson on random walk coupling

## Abstract

Exact coupling of random walks is studied. Conditions for admitting a successful exact coupling are given that are necessary and in the Abelian case also sufficient. In the Abelian case, it is shown that a random walk $S$ with step-length distribution $\mu$ started at $0$ admits a successful exact coupling with a version $S^x$ started at $x$ if and only if there is $n\geq 1$ with $\mu^{n} \wedge \mu^{n}(x+\cdot) \neq 0$. Moreover, when a successful exact coupling exists, the total variation distance between $S_n$ and $S^x_n$ is determined to be $O(n^{-1/2})$ if $x$ has infinite order, or $O(\rho^n)$ for some $\rho \in (0,1)$ if $x$ has finite order. In particular, this paper solves a problem posed by H. Thorisson on successful exact coupling of random walks on $\mathbb{R}$. It is also noted that the set of such $x$ for which a successful exact coupling can be constructed is a Borel measurable group. Lastly, the weaker notion of possible exact coupling and its relationship to successful exact coupling are studied.