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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

On ergodic sequences of measures


Authors: J. R. Blum and R. Cogburn
Journal: Proc. Amer. Math. Soc. 51 (1975), 359-365
MSC: Primary 43A05; Secondary 22D40
MathSciNet review: 0372529
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Abstract: Let $ Z$ be the group of integers and $ \bar Z$ its Bohr compactification. A sequence of probability measures $ \{ {\mu _n},n = 1,2, \ldots \} $ defined on $ Z$ is said to be ergodic provided $ {\mu _n}$ converges weakly to $ \bar \mu $, the Haar measure on $ \bar Z$. Let $ {A_n} \subset Z,n = 1,2, \ldots $ and define $ {\mu _n}$ by $ {\mu _n}(B) = \vert{A_n} \cap B\vert/\vert{A_n}\vert$ where $ \vert B\vert$ is the cardinality of $ B$. Then it is easy to show that if $ \vert{A_n} \cap {A_n} + k\vert/\vert{A_n}\vert \to 1$ for every $ k \in Z$, then $ {\mu _n}$ is ergodic. Let $ 0 \leq {p_k} \leq 1$. In this paper we construct (random) sequences $ \{ {\mu _n}\} $ which are ergodic, and such that $ \lim (\vert{A_n} \cap {A_n} + k\vert/\vert{A_n}\vert) = {p_k}$, for every $ k \in Z$.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1975-0372529-1
PII: S 0002-9939(1975)0372529-1
Article copyright: © Copyright 1975 American Mathematical Society