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

Part of: Limit theorems Numerical methods in Fourier analysis Probabilistic methods, simulation and stochastic differential equations Distribution theory - Probability

Published online by Cambridge University Press:  30 January 2018

Ankush Agarwal ,
Santanu Dey  and
Sandeep Juneja
Ankush Agarwal*
Affiliation:
Tata Institute of Fundamental Research
Santanu Dey*
Affiliation:
Tata Institute of Fundamental Research
Sandeep Juneja*
Affiliation:
Tata Institute of Fundamental Research
*
∗ Postal address: STCS, Tata Institute of Fundamental Research, 1 Homi Bhabha Road, Mumbai, India.
∗ Postal address: STCS, Tata Institute of Fundamental Research, 1 Homi Bhabha Road, Mumbai, India.
∗ Postal address: STCS, Tata Institute of Fundamental Research, 1 Homi Bhabha Road, Mumbai, India.
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Abstract

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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 queueing 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.

Keywords

Rare event simulationimportance samplingsaddle-point approximationFourier inversionlarge deviations

MSC classification

Primary: 65C05: Monte Carlo methods 60E10: Characteristic functions; other transforms 60F10: Large deviations
Secondary: 65C50: Other computational problems in probability 65T99: None of the above, but in this section
Type
Research Article
Information
Journal of Applied Probability , Volume 50 , Issue 3 , September 2013 , pp. 703 - 720
Copyright
© Applied Probability Trust 

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