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

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 $π$. 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 $σ_{n}^{>}$ of $(M_{n},S_{n})_{n\ge 0}$ are a.s. finite and the ladder chain $(M_{σ_{n}^{>}})_{n\ge 0}$ is positive recurrent on some $\mathcal{S}^{>}\subset\mathcal{S}$. We also provide simple expressions for its stationary distribution $π^{>}$, 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 $\limsup_{n\to\infty}S_{n}=\infty$ 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.