Exact Coupling of Random Walks on Polish Groups

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 >= 1 with mu(n) boolean AND mu(n)( x + center dot) not equal 0. Moreover, when a successful exact coupling exists, the total variation distance between S-n and S-n(x) is determined to be O(n(-1/2)) if x has infinite order, or O(rho(n)) for some rho is an element of (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 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.