Random walk on countably infinite Abelian groups

Abstract

Here 0 is the identi ty element of the (additive) group g~, #~0> is the probabili ty measure all of whose mass is concentrated at O, #~1~=# and #~> is the n-fold convolution of # with itself. Roughly speaking, the purpose of this paper is to imitate and extend basic results in [10] (Chapter 7 and parts of earlier chapters). There the at tention was strictly confined to the groups (~ =Z~, the groups of d-dimensional integers, or lattice points in Euclidean space of dimension d. Thus the basic ideas, methods, and notation are exactly those in [10] when possible--and most of the difficulties which arise because (~ is more complicated than Z~ can be overcome by the use of certain measures induced by the given measure tt on cyclic subgroups of (~. I t will be assumed throughout that the measure /x is aperiodic , i.e. that the support of tt genera tes all of (~. (Note however that (~ must be infinite. When @ is finite everything we do is either trivial or well known but the results are by no means the same.) Given /z we define on (~ the Markov process (random walk) X~ with transition function P z IX1 = Y] = P ( x , y) = la(y x),

Locations

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