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 <i>S</i><sub><i>M</i></sub>=<i>Z</i><sub>1</sub>+&cdots;+<i>Z</i><sub><i>M</i></sub> where the <i>Z</i><sub><i>i</i></sub>'s are independently and identically distributed (i.i.d.) random variables in R and <i>M</i> is a nonnegative, integer-valued random variable independent of the <i>Z</i><sub><i>i</i></sub>'s. We construct the first estimator that can be rigorously shown to be strongly efficient only under the assumption that the <i>Z</i><sub><i>i</i></sub>'s are subexponential and <i>M</i> 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.

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