Exact Coupling of Random Walks on Polish Groups

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 $μ$ 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 $μ^{n} \wedge μ^{n}(x+\cdot) \neq 0$. 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 $O(ρ^n)$ for some $ρ\in (0,1)$ 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.