Efficient Simulation for Large Deviation Probabilities of Heavy - Tailed Sums

Let (Xn : n ≥ 0) be a sequence of iid rv’s with mean zero and finite variance. We present an efficient statedependent importance sampling algorithm for estimating the tail of Sn = X1 + ...+Xn in a large deviations framework as n ↗ ∞. Our algorithm can be shown to be strongly efficient basically throughout the whole large deviations region as n ↗ ∞ (in particular, for probabilities of the form P (Sn > κn) as κ > 0). The techniques combine results of the theory of large deviations for sums of regularly varying distributions and the basic ideas can be applied to other rare-event simulation problems involving both light and heavy-tailed features.