Strong Ratio Limit Theorems for $\phi$-Recurrent Markov Chains

Abstract

Let $\{X_n; n = 0, 1, \cdots\}$ be a $\phi$-recurrent Markov chain on a general measurable state space $(S, \mathscr{F})$ with transition probabilities $P(x, A), x \in S, A \in \mathscr{F}$. The convergence of the ratio $\lambda P^{n+m}f / \mu P^ng$ (as $n \rightarrow \infty$), where $\lambda$ and $\mu$ are nonnegative measures on $(S, \mathscr{F})$ and $f$ and $g$ are nonnegative measurable functions on $S$, is studied. We show that the ratio converges, provided that $\lambda, \mu, f$ and $g$ are in a certain sense "small," and provided that for an embedded renewal sequence $\{u(n)\}$ the limit $\lim u(n + 1)/u(n)$ exists.