Exact Coupling of Random Walks on Polish Groups

James T. Murphy III

Published 2017-06-21Version 1

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 $x$ iff there is $n$ with $\mu^{n} \wedge \mu^{n}(x+\cdot) \neq 0$. 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 is studied.