Uniform Markov renewal theory and ruin probabilities in Markov random walks

Let {Xn,n≥0} be a Markov chain on a general state space X with transition probability P and stationary probability π. Suppose an additive component Sn takes values in the real line R and is adjoined to the chain such that {(Xn,Sn),n≥0} is a Markov random walk. In this paper, we prove a uniform Markov renewal theorem with an estimate on the rate of convergence. This result is applied to boundary crossing problems for {(Xn,Sn),n≥0}. To be more precise, for given b≥0, define the stopping time τ=τ(b)=inf {n:Sn>b}. When a drift μ of the random walk Sn is 0, we derive a one-term Edgeworth type asymptotic expansion for the first passage probabilities Pπ{τ<m} and Pπ{τ