On the Conditional Distributions and the Efficient Simulations of Exponential Integrals of Gaussian Random Fields

Details On the Conditional Distributions and the Efficient Simulations of Exponential Integrals of Gaussian Random Fields On the Conditional Distributions and the Efficient Simulations of Exponential Integrals of Gaussian Random Fields

2012-04-25

arxiv.org/abs/1204.5546v1

Mathematics

Probability

Scientific paper

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.

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