On Concentration Functions on Discrete Groups

Abstract

Let $(\xi_n)_{n \geq 0}$ be a random walk on a countable group $G$. Sufficient and necessary conditions for the existence of a finite set $A \subseteq G$ and a sequence $g_n \in G$ such that for all natural $n$ we have $P(\xi_n \in A\mid\xi_0 = g_n) = 1$ are presented. This provides a complete solution to the problem of behavior of concentration functions on discrete groups.