Uniform Markov Renewal Theory and Ruin Probabilities in Markov Random Walks

Let {X_n,n\geq0} be a Markov chain on a general state space X with transition probability P and stationary probability \pi. Suppose an additive component S_n takes values in the real line R and is adjoined to the chain such that {(X_n,S_n),n\geq0} 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 {(X_n,S_n),n\geq0}. To be more precise, for given b\geq0, define the stopping time \tau=\tau(b)=inf{n:S_n>b}. When a drift \mu of the random walk S_n is 0, we derive a one-term Edgeworth type asymptotic expansion for the first passage probabilities P_{\pi}{\tau<m} and P_{\pi}{\tau<m,S_m<c}, where m\leq\infty, c\leq b and P_{\pi} denotes the probability under the initial distribution \pi. When \mu\neq0, Brownian approximations for the first passage probabilities with correction terms are derived.