More on limit theorems for iterates of probability measures on semigroups and groups |
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Authors: | Barbara Center Arunava Mukherjea |
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Institution: | (1) University of Tampa, 33606 Tampa, Fl., USA;(2) University of South Florida, 33620 Tampa, Fl., USA |
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Abstract: | Summary In this paper, we continue earlier works of one of the authors on vague convergence of the sequence
k,n=
k+1 *...*
n, where
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
being the support of a probability measure on S. Suppose there exists an open set V with compact closure such that x
–1
Vx=V for every xS. Then for all compact sets K, sup{
n
(Kx): xS0 as n. Theorem. Let S be an at most countable discrete group. Let
n be a sequence of probability measures on S. Then for all nonnegative integers k, the sequence
k,n converges vaguely to some probability measure if and only if there exists a finite subgroup G such that the series
and for any proper subgroup G of G and any choice of elements gn in S, the series
. A sufficient condition for the vague convergence of the sequence
k,n to a probability measure is that (i) there exists a finite subgroup G such that
and (ii)
n(e)>s>0 for all n, e being the identity.The author was supported by NSF grant MCS77-03639 |
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Keywords: | |
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