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

We prove results on exact asymptotics as $n\to\infty$ for the expectations $\mathsf{E}_a \exp\bigl\{-\theta\sum_{k=0}^{n-1} g(X_k)\bigr\}$ and probabilities $\mathsf{P}_a\bigl\{\frac{1}{n}\sum_{k=0}^{n-1}g(X_k)<d\bigr\}$, where $\{\xi_k\}_{k=1}^\infty $ is a sequence of independent identically Laplace-distributed random variables, $X_n=X_0+\sum_{k=1}^n \xi_k$, $n\geqslant 1$, is the corresponding random walk on $\mathbb{R}$, $g(x)$ is a positive continuous function satisfying certain conditions, and $d>0$, $\theta>0$, $a\in\mathbb{R}$ 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 $e^{\alpha |x|}-1$, $0<\alpha<1/2$, $x\in\mathbb{R}$, for example. We give a detailed treatment of the case when $g(x)=|x|$ using Bessel functions to make explicit calculations.