Stationary entrance Markov chains, inducing, and level-crossings of random walks

TL;DR

This paper develops a framework for analyzing invariant measures of entrance and exit Markov chains derived from a base Markov chain, with applications to random walks and level-crossing phenomena.

Contribution

It introduces explicit formulas for invariant measures of entrance and exit chains under recurrence assumptions and studies their ergodic properties, extending classical results via infinite ergodic theory techniques.

Findings

Invariant measures are explicitly characterized for entrance and exit chains.

The ergodicity of the overshoot chain for oscillating random walks is established.

A central limit theorem for level crossings in zero-mean random walks is proved.

Abstract

For a Markov chain $Y$ with values in a Polish space, consider the entrance Markov chain obtained by sampling $Y$ at the moments when it enters a fixed set $A$ from its complement $A^c$. Similarly, consider the exit Markov chain, obtained by sampling $Y$ at the exit times from $A^c$ to $A$. This paper provides a framework for analysing invariant measures of these two types of Markov chains in the case when the initial chain $Y$ has a known $\sigma$-finite invariant measure. Under certain recurrence-type assumptions ($Y$ can be transient), we give explicit formulas for invariant measures of these chains. Then we study their uniqueness and ergodicity assuming that $Y$ is topologically recurrent, irreducible, and weak Feller. Our approach is based on the technique of inducing from infinite ergodic theory. This also yields, in a natural way, the versions of the results above (provided in the paper) for the classical induced Markov chains. 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 for random walks with zero mean and finite variance of increments.