Let {(Xn , S n ), n ≥ 0} be a Markov random walk in which X n takes values in a general state space and S n takes values on the real line R. In this paper we present some results that are useful in the study of asymptotic approximations of boundary crossing problems for Markov random walks. The main results are asymptotic expansions on moments of the first ladder height in Markov random walks with small positive drift. In order to establish the asymptotic expansions we study a uniform Markov renewal theorem, which relates to the rate of convergence for the distribution of overshoot, and present an analysis of the covariance between the first passage time and the overshoot.

Let (X, S) = {(Xn , Sn ); n ≧0} be a Markov random walk with finite state space. For a ≦ 0 < b define the stopping times τ= inf {n:Sn > b} and T= inf{n:Sn ∉(a, b)}. The diffusion approximations of a one-barrier probability P {τ < ∝ | X o= i}, and a two-barrier probability P{ST ≧b | X o = i} with correction terms are derived. Furthermore, to approximate the above ruin probabilities, the limiting distributions of overshoot for a driftless Markov random walk are involved.