Stationarity of entrance Markov chains and overshoots of random walks

Consider the Markov chain of overshoots over the zero level of a one-dimensional random walk S that oscillates between−∞and +∞. We establish an explicit formula for an invariant measure of this chain and prove its uniqueness and ergodicity. The main corollary is a limit theorem for the number of level crossings on a large time interval for walks with zero mean and finite variance of increments. In the context of these results about the invariant measure, it is natural to consider the random walk S as a “stationary” Markov chain starting under the Lebesgue measure on R and use the technique of inducing from infinite ergodic theory. We develop this approach in a general setting and apply it to the chain of overshoots, which is a particular case of the entrance Markov chain obtained by sampling an arbitrary Markov chain Y (in a Polish space, and possibly transient) at the moments when Y enters a fixed set A from its complement Ac. In addition, we consider the exit and induced Markov chains, obtained by sampling Y at the exit times from Ac into A and restricting Y to A, respectively. This paper provides a framework for analysing invariant measures of such entrance, exit, and induced chains in the case when the initial chain Y has a known invariant measure. We find invariant measures explicitly, and then study their uniqueness (and ergodicity) assuming that the chain Y is topologically recurrent and weak Feller.