Efficient simulation of density and probability of large deviations of sum of random vectors using saddle point representations

TL;DR

This paper introduces a saddle-point based importance sampling method for efficiently estimating the density and large deviation probabilities of sums of i.i.d. light-tailed vectors, improving accuracy and computational efficiency.

Contribution

It develops a novel saddle-point representation approach that enables identification of zero-variance importance sampling measures with asymptotically vanishing relative error.

Findings

The proposed method achieves asymptotically vanishing relative error.

It effectively estimates tail densities and large deviation probabilities.

Extension to expected overshoot estimation demonstrates broader applicability.

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 queueing and financial credit risk modelling. 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.