Rare Events Simulation for Heavy-Tailed Distributions

Abstract

This paper studies rare events simulation for the heavy--tailed case, where some of the underlying distributions fail to have the exponential moments required for the standard algorithms for the light--tailed case. Several counterexamples are given to indicate that in the heavy--tailed case, there are severe problems with the approach of developing limit results for the conditional distribution given the rare event and use this as basis for importance sampling. On the positive side, two algorithms having a relative error which is almost bounded are presented, one based upon order statistics and the other upon a different importance sampling idea. Keywords and phrases conditional Monte Carlo, importance sampling, large deviations, logarithmic efficiency, M/G/1 queue, order statistics, random walk, regular variation, subexponential distribution 1 Introduction Estimation of small probabilities by simulation is one of the key issues of todays simulation literature. The reason is two--fol...

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