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

Let $\{\xi_k\}_{k=0}^\infty$ 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 $\mathbf P\{\frac1n\sum_{k=0}^{n-1}g(\xi_k)<d\}$, $n\to\infty$, 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 $x\ge0$.