Asymptotic expansions in multidimensional Markov renewal theory and first passage times for Markov random walks

We prove a d-dimensional renewal theorem, with an estimate on the rate of c onvergence, for Markov random walks. This result is applied to a variety of boundary crossing problems for a Markov random walk {(X-n, S-n), n greater than or equal to 0}, in which X-n takes values in a general state space an d S-n takes values in R-d. In particular, for the case d = 1, we use this r esult to derive an asymptotic formula for the variance of the first passage time when S-n exceeds a high threshold b, generalizing Smith's classical f ormula in the case of i.i.d. positive increments for S-n. For d > 1, we app ly this result to derive an asymptotic expansion of the distribution of (X- T, S-T), where T = inf {n : S-n,S-1 > b} and S-n,S-1 denotes the first comp onent of S-n.