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

Abstract

We prove a d-dimensional renewal theorem, with an estimate on the rate of convergence, 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 ≥0, in which X n takes values in a general state space and S n takes values in ℝ d . In particular, for the case d = 1, we use this result 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 formula in the case of i.i.d. positive increments for S n . For d > 1, we apply this result to derive an asymptotic expansion of the distribution of (X T ,S T ), where T = inf { n : S n,1 > b } and S n,1 denotes the first component of S n .