## Journal of Applied Probability

- J. Appl. Probab.
- Volume 50, Number 3 (2013), 703-720.

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

Ankush Agarwal, Santanu Dey, and Sandeep Juneja

#### 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 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.

#### Article information

**Source**

J. Appl. Probab., Volume 50, Number 3 (2013), 703-720.

**Dates**

First available in Project Euclid: 5 September 2013

**Permanent link to this document**

https://projecteuclid.org/euclid.jap/1378401231

**Digital Object Identifier**

doi:10.1239/jap/1378401231

**Mathematical Reviews number (MathSciNet)**

MR3102510

**Zentralblatt MATH identifier**

1282.65025

**Subjects**

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

**Keywords**

Rare event simulation importance sampling saddle-point approximation Fourier inversion large deviations

#### Citation

Agarwal, Ankush; Dey, Santanu; Juneja, Sandeep. Efficient simulation of large deviation events for sums of random vectors using saddle-point representations. J. Appl. Probab. 50 (2013), no. 3, 703--720. doi:10.1239/jap/1378401231. https://projecteuclid.org/euclid.jap/1378401231