Nonlinear renewal theory for Markov random walks

AbstractLet {Sn} be a Markov random walk satisfying the conditions of Kesten's Markov renewal theorem. It is shown that if {Zn} is a stochastic process whose finite-dimensional, conditional distributions are asymptotically close to those of {Sn} (in the sense of weak convergence), then the overshoot of {Zn} has the same limiting distribution as that of {Sn}. In the case where {Zn} can be represented as a perturbed Markov random walk, this allows substantial weakening of the slow change condition on the perturbation process; more importantly, no such representation is required. An application to machine breakdown times is given