Large deviations for distributions of sums of random variables: Markov chain method

Let {ξk}k=0∞ be a sequence of i.i.d. real-valued random variables, and let g(x) be a continuous positive function. Under rather general conditions, we prove results on sharp asymptotics of the probabilities \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ P\left\{ {\frac{1} {n}\sum\limits_{k = 0}^{n - 1} {g\left( {\xi _k } \right) < d} } \right\} $$\end{document}, n → ∞, and also of their conditional versions. The results are obtained using a new method developed in the paper, namely, the Laplace method for sojourn times of discrete-time Markov chains. We consider two examples: standard Gaussian random variables with g(x) = |x|p, p > 0, and exponential random variables with g(x) = x for x ≥ 0.