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.
Citation
Jingchen Liu. Gongjun Xu. "On the conditional distributions and the efficient simulations of exponential integrals of Gaussian random fields." Ann. Appl. Probab. 24 (4) 1691 - 1738, August 2014. https://doi.org/10.1214/13-AAP960
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