  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.
