Recurrence time and range for a generalized random walk

Abstract

The notions of recurrence time, range, and the limit of probabilities P k of return to the origin arise in the study of random walks on groups. We examine these notions and develop relationships among them in an ergodic theory setting in which the usual requirement of independence of the increments of the random walk can be relaxed to simply an ergodic requirement. Thus we consider generalized random walks or GRWs. The ergodic theory setting is related to Mackey's virtual group theory in that the GRW determines a virtual group homomorphism (or cocycle). We relate the condition- that the homomorphism is trivial (the cocycle is a coboundary) to the Cesáro limit of P k . The basic ideas of virtual group theory were established by Mackey and further developed by Ramsay. Our virtual group homomorphism result does not require familiarity with the technicalities of virtual group theory.