A Lower Bound of the Asymptotic Behavior of Some Markov Processes

Abstract

Let $X_0, X_1 \cdots$ be a Markov process with transition function $p(x, dy)$. Let $L_n(\omega, \cdot)$ be its average occupation time measure, i.e., $L_n(\omega, A)= 1/n \cdot \sum^{n-1}_{i=0} \chi A(x_i(\omega))$. A powerful theorem concerning the lower bound of the asymptotic behavior of $L_n(\omega, \cdot)$ was proved by Donsker and Varadhan when $p(x, dy)$ satisfies a homogeneity condition. This paper tries to extend their results to some cases where such a homogeneity condition is not satisfied. This particularly includes symmetric random walks and Harris' chains.