Efficient Simulation for Large Deviation Probabilities of Sums of Heavy-Tailed Increments

Let (X
_n
:n ges 0) be a sequence of iid rv's with mean zero and finite variance. We describe an efficient state-dependent importance sampling algorithm for estimating the tail of S
_n
= X
₁
+ ... + X
_n
in a large deviations framework as n - infin. Our algorithm can be shown to be strongly efficient basically throughout the whole large deviations region as n - infin (in particular, for probabilities of the form P (S
_n
> kn) as k > 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