Ladder epochs and ladder chain of a Markov random walk with discrete driving chain

Abstract

Let $(M_n,S_n{)}_{n\ge 0}$ be a Markov random walk with positive recurrent driving chain $(M_n{)}_{n\ge 0}$ having countable state space $\mathcal{S}$ and stationary distribution $\pi$. It is shown in this note that, if the dual sequence $({}^{\#}{M}_n,{}^{\#}{S}_n{)}_{n\ge 0}$ is positive divergent, i.e. ${}^{\#}{S}_n\to \infty$ a.s., then the strictly ascending ladder epochs ${\sigma }_n^{>}$ of $(M_n,S_n{)}_{n\ge 0}$ are a.s. finite and the ladder chain $(M_{{\sigma }_n^{>}}{)}_{n\ge 0}$ is positive recurrent on some ${\mathcal{S}}^{>}\subset \mathcal{S}$. We also provide simple expressions for its stationary distribution ${\pi }^{>}$, an extension of the result to the case when $(M_n{)}_{n\ge 0}$ is null recurrent, and a counterexample that demonstrates that ${}^{\#}{S}_n\to \infty$ a.s. does not necessarily entail $S_n\to \infty$ a.s., but rather ${\mathrm{limsup}}_{n\to \infty }{S}_n=\infty$ a.s. only. Our arguments are based on Palm duality theory, coupling and the Wiener-Hopf factorization for Markov random walks with discrete driving chain.