On Monte Carlo estimation of large deviations probabilities

Abstract

Importance sampling is a Monte Carlo technique where random data are sampled from an alternative "sampling distribution" and an unbiased estimator is obtained by likelihood ratio weighting. Here we consider estimation of large deviations probabilities via importance sampling. Previous works have shown, for certain special cases, that "exponentially twisted" distributions possess a strong asymptotic optimality property as a sampling distribution. The results of this paper unify and generalize the previous special case results. The analysis is presented in an abstract setting, so the results are quite general and directly applicable to a number of large deviations problems. Our main motivation, however, is to attack sample path problems. To illustrate the application to this class of problems, we consider Mogulskii type sample path problems in some detail.