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

We prove ad-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 (Xn,Sn),n≥0, in whichXntakes values in a general state space andSntakes values in ℝd. In particular, for the cased= 1, we use this result to derive an asymptotic formula for the variance of the first passage time whenSnexceeds a high thresholdb, generalizing Smith's classical formula in the case of i.i.d. positive increments forSn. Ford> 1, we apply this result to derive an asymptotic expansion of the distribution of (XT,ST), whereT= inf {n:Sn,1>b} andSn,1denotes the first component ofSn.