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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

On recurrent random walks on semigroups


Authors: T. C. Sun, A. Mukherjea and N. A. Tserpes
Journal: Trans. Amer. Math. Soc. 185 (1973), 213-227
MSC: Primary 60J15
DOI: https://doi.org/10.1090/S0002-9947-1973-0346916-4
MathSciNet review: 0346916
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Abstract: Let $ \mu $ be a regular Borel probability measure on a locally compact semigroup S and consider the right (resp. left) random walk on $ D = \overline {{\text{U}}{F^n}} ,F = {\text{Supp}}\;\mu $, with transition function $ {P^n}(x,B) \equiv {\mu ^n}({x^{ - 1}}B)\;({\text{resp}}.\;{\mu ^n}(B{x^{ - 1}}))$. These Markov chains can be represented as $ {Z_n} = {X_1}{X_2} \cdots {X_n}\;({\text{resp}}.\;{S_n} = {X_n}{X_{n - 1}} \cdots {X_1}),\;{X_i}$'s independent $ \mu $-distributed with values in S defined on an infinite-sequence space $ (\Pi _1^\infty {S_i},P),{S_i} = S$ for all i. Let $ {R_r}\;({\text{resp}}.\;{R_t}) = \{ x \in D;{P_x}({Z_n}({S_n}) \in {N_x}\;{\text{i.o.}}) = 1$ for all neighborhoods $ {N_x}$ of x} and $ {R'_r}({R'_t}) = \{ x \in D;P({Z_n}({S_n}) \in {N_x}\;{\text{i.o.}}) = 1$ for all $ {N_x}$ of x}. Let S be completely simple ( $ = E \times G \times F$, usual Rees product) in the results (1), (2), (3), (4), (5) below: (1) $ x \in {R_r}\;iff\;\Sigma \;{\mu ^n}({x^{ - 1}}{N_x}) = \infty $ for all neighborhoods $ {N_x}$ of $ x\;iff\;\Sigma \;{\mu ^n}({N_x}) = \infty $ for all $ {N_x}$ of x. (2) Either $ {R_r} = {R_t} = \emptyset $ or $ {R_r} = {R_t} = D = $ also completely simple. (3) If the group factor G is compact, then there are recurrent values and we have $ {R_r} = {R_t} = D = $ completely simple. (4) $ {R'_r} \ne \emptyset $ implies $ {R'_r} = {R_r} = {R_t} = D = $ a right subgroup of S (but $ {R'_t}$ may be $ \emptyset $). (5) S can support a recurrent random walk (i.e., a r. walk with $ {R_r} \ne \emptyset $) iff G (= the group factor) can support a recurrent random walk. Finally (6) if S is compact abelian, then always $ R' = R = K = $ the kernel of S. These results extend previously known results of Chung and Fuchs and Loynes.


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DOI: https://doi.org/10.1090/S0002-9947-1973-0346916-4
Keywords: Sums of independent random variables with values in a semigroup, stationary recurrent random walks, completely simple topological semigroups
Article copyright: © Copyright 1973 American Mathematical Society