Efficient Rare Event Simulation for Heavy-Tailed Compound Sums

We develop an efficient importance sampling algorithm for estimating the tail distribution of heavy-tailed compound sums, that is, random variables of the form S-M = Z(1) + . . . + Z(M) where the Z(i)'s are independently and identically distributed (i.i.d.) random variables in R and M is a nonnegative, integer-valued random variable independent of the Z(i)'s. We construct the first estimator that can be rigorously shown to be strongly efficient only under the assumption that the Z(i)'s are subexponential and M is light-tailed. Our estimator is based on state-dependent importance sampling and we use Lyapunov-type inequalities to control its second moment. The performance of our estimator is empirically illustrated in various instances involving popular heavy-tailed models.