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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Random walks on compact semigroups


Authors: A. Mukherjea, T. C. Sun and N. A. Tserpes
Journal: Proc. Amer. Math. Soc. 39 (1973), 599-605
MSC: Primary 60J15; Secondary 60B15
MathSciNet review: 0317420
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Abstract: Let $ \beta $ be a regular Borel probability measure with support $ F$ on a compact semigroup $ S$. Let $ {X_1},{X_2}, \cdots $ be a sequence of independent random variables on some probability space $ (\Omega ,\Sigma ,P)$ with values in $ S$, having identical distribution $ P({X_n} \in B) = \beta (B)$. The random walk $ {Z_n} = {X_1}{X_2} \cdots {X_n}$ is studied in this paper. Let $ D$ be the closed semigroup generated by $ F$. An element $ x$ in $ D$ is called recurrent iff $ {P_x}({Z_n} \in {N_x}i.o.) = 1$ for every open set $ {N_x}$ containing $ x$. This paper characterizes the recurrence of an element $ x$ in terms of divergence of the series $ \sum\nolimits_{n = 1}^\infty {{\beta ^n}} ({N_x})$ for every open set $ {N_x}$ containing $ x$. It also shows that the set of recurrent states of $ \{ {Z_n}\} $ is precisely the kernel of $ D$.


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DOI: https://doi.org/10.1090/S0002-9939-1973-0317420-X
Keywords: Topological semigroup, kernel of a semigroup, recurrent random walks
Article copyright: © Copyright 1973 American Mathematical Society