Stationary entrance chains and applications to random walks

Abstract

For a Markov chain $Y$ with values in a Polish space, consider the entrance chain obtained by sampling $Y$ at the moments when it enters a fixed set $A$ from its complement $A^c$. Similarly, consider the exit chain, obtained by sampling $Y$ at the exit times from $A^c$ to $A$. We use the method of inducing from ergodic theory to study invariant measures of these two types of Markov chains in the case when the initial chain $Y$ has a known invariant measure. We give explicit formulas for invariant measures of the entrance and exit chains under certain recurrence-type assumptions on $A$ and $A^c$, which apply even for transient chains. Then we study uniqueness and ergodicity of these invariant measures assuming that $Y$ is topologically recurrent, topologically irreducible, and weak Feller.

We give applications to random walks in $R^d$, which we regard as “stationary” Markov chains started under the Lebesgue measure. We are mostly interested in dimension one, where we study the Markov chain of overshoots above the zero level of a random walk that oscillates between $-\infty$ and $+\infty$. We show that this chain is ergodic, and use this result to prove a central limit theorem for the number of level crossings of a random walk with zero mean and finite variance of increments.