Sums of dependent random variables

Abstract

We obtain limit theorems for certain probabilities and expected values related to sums of identically distributed random variables with values in an amenable unimodular group, A (additive notation is used although A is not necessarily commutative). Our results apply to the case of independent random variables (the random walk case) but are not quite as good as the usual random walk results, since our limits are obtained in the generalized Cesiro sense, i.e., limits of the average over a set A,, in a summing sequence for A. Accordingly, the main advantage for our methods (which come from the development of the theory of Mackey’s virtual groups) is that they work well in the case of dependent random variables. The sums of random variables define generalized random walks (GRWs), just as random walks are defined in the independent case. The use of Cesiro type convergence makes unnecessary the distinction between periodic and aperiodic random walks. Unless A is countable, our approach is not able to formulate the condition that the GRW S at step k, S,, is in a Bore1 subset E of A-instead we measure the overlap of D + S, and E, where D can be taken to be a small neighborhood of 0 in A. Our results are mostly twosided, i.e., concerned with the values attained by a GRW walking backward or forward. The basic material on Mackey’s virtual groups is established in (81 and [9]. However, our range closure homomorphism of Theorem 3Sa requires Ramsay’s more complicated definition of a homomorphism in [ lOJ, unless A is countable. Most of our results do not require familiarity with the technicalities of virtual group theory. To describe transient GRWs we will introduce two parameters, r(S; E) in Theorem 2.4, which describes the range of values reached by S, and k(S; E) in Definition 3.3, which describes how often E is sent to E-the kernel of S. The product of these two parameters leads to Z(S), which is of significance in the theory of virtual group homomorphisms. We obtain rather specific results regarding k(S; E) and r(S; E) in the case where A is the real numbers and S, > 0. For transient S we obtain a way of 120 0022-247X/80/090120-12$02.00/0