Rare events simulation for heavy-tailed distributions

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; this is used as a 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.