Let {(Xn, Sn), n ≥ 0} be a Markov random walk in which Xn takes values in a general state space and Sn 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.

A neural model with N interacting neurons is considered. A firing of neuron i delays the firing times of all other neurons by the same random variable θ(i), and in isolation the firings of the neuron occur according to a renewal process with generic interarrival time Y(i). The stationary distribution of the N-vector of inhibitions at a firing time is computed, and involves waiting distributions of GI/G/1 queues and ladder height renewal processes. Further, the distribution of the period of activity of a neuron is studied for the symmetric case where θ(i) and Y(i) do not depend upon i. The tools are probabilistic and involve path decompositions, Palm theory and random walks.

Previous work in extending Wald's equations to Markov random walks involves finiteness of moment generating functions and uniform recurrence assumptions. By using a new approach, we can remove these assumptions. The results are applied to establish finiteness of moments of ladder variables and to derive asymptotic expansions for expected first passage times of Markov random walks. Wiener–Hopf factorizations for Markov random walks are also applied to analyse ladder variables and related first passage problems.

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 < ∝ | Xo= i}, and a two-barrier probability P{ST ≧b | Xo = 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.