Exact asymptotics of probabilities of large deviations for Markov chains: the Laplace method

Abstract

We prove results on exact asymptotics as for the expectations and probabilities , where is a sequence of independent identically Laplace-distributed random variables, , , is the corresponding random walk on , is a positive continuous function satisfying certain conditions, and , , are fixed numbers. Our results are obtained using a new method which is developed in this paper: the Laplace method for the occupation time of discrete-time Markov chains. For one can take , , , , or , , , for example. We give a detailed treatment of the case when using Bessel functions to make explicit calculations.