More on limit theorems for iterates of probability measures on semigroups and groups

Abstract

In this paper, we continue earlier works of one of the authors on vague convergence of the sequence βk,n=βk+1 *...*βn, where βn is a sequence of probability measures on semigroups or groups. Typical results in this paper are: Theorem. Let S be a locally compact noncompact second countable group such that \(S = \overline {\bigcup\limits_{n = 1}^\infty {S_\beta ^n ,} } S_\beta\) being the support of a probability measure β on S. Suppose there exists an open set V with compact closure such that x−1Vx=V for every x∈S. Then for all compact sets K, sup{βn(Kx): x∈S→0 as n→∞. Theorem. Let S be an at most countable discrete group. Let βn be a sequence of probability measures on S. Then for all nonnegative integers k, the sequence βk,n converges vaguely to some probability measure if and only if there exists a finite subgroup G such that the series \(\sum\limits_{n = 1}^\infty {\beta _n } (S - G) s>0 for all n, e being the identity.