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

We prove results on exact asymptotics as n -> infinity for the expectations E-a exp{-theta Sigma(n-1)(k=0) g(X-k)} and probabilities Pa{1/n Sigma(n-1)(k=0) g(X-k) <d}, where {xi(k)}(k-1)(infinity) is a sequence of independent identically Laplace-distributed random variables, X-n - X-0 + Sigma(n)(k=1) xi(k), n >= 1, is the corresponding random walk on R, g(x) is a positive continuous function satisfying certain conditions, and d > 0, theta > 0, a is an element of R 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 g(x) one can take |x|(p), log(|x|(p) + 1), p > 0, |x| log(|x|+1), or e(alpha|x|)-1, 0 < alpha < 1/2, x is an element of R, for example. We give a detailed treatment of the case when g(x) = |x| using Bessel functions to make explicit calculations.