On the Range of Recurrent Markov Chains

Abstract

Let $R_n$ be the number of distinct elements among $X_0, X_1,\cdots, X_n$, where $\{X_n\}$ is an irreducible recurrent Markov chain. It is shown that, under an appropriate condition, $n^{-1}R_n \rightarrow 0$ a.s. $(P_a)$ where $a$ is any state and $P_a$ is conditional probability measure given $X_0 = a$. We prove that any recurrent random walk satisfies our condition, so that the result contains the well-known random walk case. We also give an example of an irreducible recurrent chain for which the result fails to hold.