## The Annals of Applied Probability

### On the conditional distributions and the efficient simulations of exponential integrals of Gaussian random fields

#### Abstract

In this paper, we consider the extreme behavior of a Gaussian random field $f(t)$ living on a compact set $T$. In particular, we are interested in tail events associated with the integral $\int_{T}e^{f(t)}\,dt$. We construct a (non-Gaussian) random field whose distribution can be explicitly stated. This field approximates the conditional Gaussian random field $f$ (given that $\int_{T}e^{f(t)}\,dt$ exceeds a large value) in total variation. Based on this approximation, we show that the tail event of $\int_{T}e^{f(t)}\,dt$ is asymptotically equivalent to the tail event of $\sup_{T}\gamma(t)$ where $\gamma(t)$ is a Gaussian process and it is an affine function of $f(t)$ and its derivative field. In addition to the asymptotic description of the conditional field, we construct an efficient Monte Carlo estimator that runs in polynomial time of $\log b$ to compute the probability $P(\int_{T}e^{f(t)}\,dt>b)$ with a prescribed relative accuracy.

#### Article information

Source
Ann. Appl. Probab. Volume 24, Number 4 (2014), 1691-1738.

Dates
First available in Project Euclid: 14 May 2014

https://projecteuclid.org/euclid.aoap/1400073661

Digital Object Identifier
doi:10.1214/13-AAP960

Mathematical Reviews number (MathSciNet)
MR3211008

Zentralblatt MATH identifier
1311.65007

Subjects
Primary: 60G15: Gaussian processes 65C05: Monte Carlo methods

#### Citation

Liu, Jingchen; Xu, Gongjun. On the conditional distributions and the efficient simulations of exponential integrals of Gaussian random fields. Ann. Appl. Probab. 24 (2014), no. 4, 1691--1738. doi:10.1214/13-AAP960. https://projecteuclid.org/euclid.aoap/1400073661

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#### Supplemental materials

• Supplementary material: Supplement to “On the conditional distributions and the efficient simulations of exponential integrals of gaussian random fields”. Proofs of Proposition 14 and Lemmas 17, 18, 20, 22, 23 and 24 are provided in the supplementary material.