Johan Ericsson: Large Deviations and Weak Convergence of Measures, with applications to Monte Carlo Estimators

Abstract.

In this thesis we apply the large deviations principle to study the performance of Monte Carlo estimators for rare events. We introduce weak convergence of measures and study the topological structure of the collection of finite signed measures in the weak topology and the in the tau-topology. We prove the law of large numbers for the empirical distributions of the importance sampling estimator and Sanov’s Theorem in the tau-topology for random variables taking values in a Polish space and then more generally in a measurable space. We also introduce a version of Sanov’s Theorem for the empirical distributions of importance sampling estimators. This is used to study the performance of importance sampling and crude Monte Carlo estimators.