Summary
In this paper, we continue earlier works of one of the authors on vague convergence of the sequence β k,n=β k+1 *...*β n, where β n 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 −1 Vx=V for every x∈S. Then for all compact sets K, sup{β n(Kx): x∈S→0 as n→∞. Theorem. Let S be an at most countable discrete group. Let β n be a sequence of probability measures on S. Then for all nonnegative integers k, the sequence β k,n converges vaguely to some probability measure if and only if there exists a finite subgroup G such that the series \(\sum\limits_{n = 1}^\infty {\beta _n } (S - G) < \infty\) and for any proper subgroup G′ of G and any choice of elements gn in S, the series \(\sum\limits_{n = 1}^\infty {\beta _n } (S - g_{n - 1} G' g_n^{ - 1} ) = \infty\). A sufficient condition for the vague convergence of the sequence β k,n to a probability measure is that (i) there exists a finite subgroup G such that \(\sum\limits_{n = 1}^\infty {\beta _n } (S - G) < \infty\) and (ii) β n(e)>s>0 for all n, e being the identity.
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The author was supported by NSF grant MCS77-03639
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Center, B., Mukherjea, A. More on limit theorems for iterates of probability measures on semigroups and groups. Z. Wahrscheinlichkeitstheorie verw Gebiete 46, 259–275 (1979). https://doi.org/10.1007/BF00538114
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DOI: https://doi.org/10.1007/BF00538114