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

Let {ξ _k } _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 $ Pleft{ {frac{1} {n}sumlimits_{k = 0}^{n - 1} {gleft( {xi _k } right) < d} } right} $ Pleft{ {frac{1} {n}sumlimits_{k = 0}^{n - 1} {gleft( {xi _k } right) < d} } right} , 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.