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.

Published in at http://dx.doi.org/10.1214/13-AAP960 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)

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• Validity of the expected Euler characteristic heuristic
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• On the distribution of the maximum of a gaussian field with d parameters
• Tail approximations of integrals of Gaussian random fields
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