On large deviations theory and asymptotically efficient Monte Carlo estimation

Importance sampling is a Monte Carlo simulation technique which uses a biased simulation distribution. In various situations, several candidate simulation densities have been proposed, but most of these have been ad hoc. The simulation design problem is considered using large deviations theory. Simulation distributions which consist of convex combinations of exponentially twisted distributions are considered. A sequence of simulation distribution is asymptotically efficient if, as some parameter n increases, approaching the limit ≐ the number of simulations required to obtain a specified estimator precision does not grow exponentially fast. The main theorem gives a sufficient condition and a necessary condition for asymptotic efficiency of the candidate simulation distributions. This is done in the multidimensional setting required by many practical simulation problems. To illustrate these ideas, simple example typical of a nonlinear communications channel is presented