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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 π. 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 τ=τ(b)=inf{n:S_n>b}. When a drift μof the random walk S_n is 0, we derive a one-term Edgeworth type asymptotic expansion for the first passage probabilities P_π{τ<m} and P_π{τ<m,S_m<c}, where m\leq\infty, c\leq b and P_π denotes the probability under the initial distribution π. When μ\neq0, Brownian approximations for the first passage probabilities with correction terms are derived.