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Bilateral random walks on compact semigroups

Authors: A. Mukherjea and N. A. Tserpes
Journal: Proc. Amer. Math. Soc. 47 (1975), 457-466
MSC: Primary 60B99
MathSciNet review: 0423459
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Abstract: Let $ \mu $ be a regular Borel probability measure with support $ F$ on a compact semigroup $ S$. Let $ {X_0},{X_{ \pm 1}},{X_{ \pm 2}}, \cdots $ be a sequence of independent random variables with values in $ S$, having identical distribution $ P({X_n} \in B) = \mu (B)$. The random walk $ {W_n} = {X_{ - n}} \cdots {X_{ - 1}}{X_0}{X_1} \cdots {X_n}$ is studied in this paper. Let $ D$ be the closed semigroup generated by $ F$ and let $ K$ be the kernel of $ D$. An element $ x \in D$ is called recurrent iff $ {P_x}({W_n} \in {N_x}{\text{i}}{\text{.o}}{\text{.}}) = 1$ for every open neighborhood $ {N_x}$ of $ x$. We prove: $ x$ is essential for $ {W_n}$ if and only if $ x \in K$ if and only if $ x$ is recurrent if and only if $ \Sigma {P_x}({W_n} \in {N_x}) = \Sigma {\mu ^n} \ast [{\mu ^n}({x^{ - 1}} \cdot )]({N_x}) = \infty $ for every $ {N_x}$. Moreover all states in $ K$ are recurrent positive. These results extend results of the authors for the unilateral random walks (using different methods) and recent results of Larisse for the discrete case.

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Keywords: Compact topological semigroup, kernel of a compact semigroup, bilateral and unilateral recurrent random walks on semigroups induced by probability measures
Article copyright: © Copyright 1975 American Mathematical Society

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