The Occupation Time of a Set by Countably Many Recurrent Random Walks

Abstract

Let $A_0(x), x \in Z$ be independent nonnegative integer-valued random variables with $\mu_j(x) = E(A_0(x)(A_0(x) - 1)\cdots(A_0(x) - j + 1))$. Assume that $\{\mu j(x)\}_x$ has limits for $j = 1,2$ and that it is bounded for $3 \leqq j \leqq 6$. Suppose at time zero there are $A_0(x)$ particles at $x \in Z$ and subsequently the particles move independently according to the transition function $P(x, y)$ of a recurrent random walk. For a finite nonempty subset $B$ of $Z$ denote by $A_n(B)$ the number of particles in $B$ at time $n$. Then $S_n(B) = \sum^n_{k=1} A_k(B)$ is the total occupation time of $B$ by time $n$ of all particles. Assuming that the $n$ step transition function $P_n(x, y)$ is such that there is an $\alpha$, with $1 < \alpha \leqq 2$, so that $P_n(0, x) \sim cn^{-1/\alpha}$ for all $x$, it is proved that the strong law of large numbers and the central limit theorem hold for the sequence $\{S_n(B)\}$.