Efficient simulation of large deviation events for sums of random vectors using saddle-point representations

Agarwal, A. , Dey, S. and Juneja, S. (2013) Efficient simulation of large deviation events for sums of random vectors using saddle-point representations. Journal of Applied Probability, 50(3), pp. 703-720. (doi: 10.1017/s0021900200009797)

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 (i.i.d.), light-tailed and nonlattice 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 the 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. Furthermore, 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 develop an asymptotically vanishing relative error estimator for the practically important expected overshoot of sums of i.i.d. random variables.

• Item Type: Articles
• Status: Published
• Refereed: Yes
• Glasgow Author(s) Enlighten ID: Agarwal, Dr Ankush
• Authors: Agarwal, A., Dey, S., and Juneja, S.
• Subjects: Q Science > QA Mathematics
• College/School: College of Social Sciences > Adam Smith Business School > Economics
• Journal Name: Journal of Applied Probability
• Journal Abbr.: JAP
• Publisher: Cambridge University Press
• ISSN: 0021-9002
• ISSN (Online): 1475-6072
• Copyright Holders: Copyright © 2013 Applied Probability Trust
• First Published: First published in Journal of Applied Probability 50(3): 703-320
• Publisher Policy: Reproduced in accordance with the publisher copyright policy