Efficient Simulation of Large Deviation Events for Sums of Random Vectors Using Saddle-Point Representations

Abstract

We consider the problem of efficient simulation estimation of the density function at the tails, and the probability of large deviations for a sum of independent, identically distributed, light-tailed and non-lattice random vectors. The latter problem besides being of independent interest, also forms a building block for more complex rare event problems that arise, for instance, in queuing and financial credit risk modeling. It has been extensively studied in literature where state independent exponential twisting based importance sampling has been shown to be asymptotically efficient and a more nuanced state dependent exponential twisting has been shown to have a stronger bounded relative error property. We exploit the saddle-point based representations that exist for these rare quantities, which rely on inverting the characteristic functions of the underlying random vectors. These representations reduce the rare event estimation problem to evaluating certain integrals, which may via importance sampling be represented as expectations. Further, it is easy to identify and approximate the zero-variance importance sampling distribution to estimate these integrals. We identify such importance sampling measures and show that they possess the asymptotically vanishing relative error property that is stronger than the bounded relative error property. To illustrate the broader applicability of the proposed methodology, we extend it to similarly efficiently estimate the practically important expected overshoot of sums of iid random variables.

State-independent Importance Sampling for Random Walks with Regularly Varying Increments

“…Siegmund [39] provides the first weakly efficient importance sampling algorithm for estimating the level crossing probabilities when the increments X n are light-tailed using large deviations based exponentially twisted change of measure. In [38], Sadowsky and Bucklew develop a weakly efficient algorithm for estimating P(S n > na) for a > EX, and X light-tailed, again using exponential twisting based importance sampling distribution (also see [37,23,12,21,2] for related analysis). This problem is important mainly because it forms a building block to many more complex rare event problems involving combination of renewal processes: for examples in queueing, see [35] and in financial credit risk modeling, see [27] and [9].…”

“…2 is the variance operator associated with measure P 1 (•) and CV (Z) = Var 1 [Z]/z is the coefficient of variation of Z.…”

State-independent importance sampling for random walks with regularly varying increments

“…Siegmund [39] provides the first weakly efficient importance sampling algorithm for estimating the level crossing probabilities when the increments X n are light-tailed using large deviations based exponentially twisted change of measure. In [38], Sadowsky and Bucklew develop a weakly efficient algorithm for estimating P(S n > na) for a > EX, and X light-tailed, again using exponential twisting based importance sampling distribution (also see [37,23,12,21,2] for related analysis). This problem is important mainly because it forms a building block to many more complex rare event problems involving combination of renewal processes: for examples in queueing, see [35] and in financial credit risk modeling, see [27] and [9].…”