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

TL;DR

This paper analyzes the tail behavior of exponential integrals of Gaussian fields, providing an explicit approximation of the conditional distribution and an efficient Monte Carlo method for probability estimation.

Contribution

It introduces a new non-Gaussian field approximation for the conditional Gaussian field and develops a polynomial-time Monte Carlo estimator for tail probabilities.

Findings

Asymptotic equivalence between tail events of the integral and the supremum of a related Gaussian process.

Explicit construction of a non-Gaussian approximation to the conditional Gaussian field.

Efficient polynomial-time Monte Carlo estimator for tail probability computation.

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 $\int_Te^{f(t)}\,dt$. We construct a (non-Gaussian) random field whose distribution can be explicitly stated. This field approximates the conditional Gaussian random field $f$ (given that $\int_Te^{f(t)}\,dt$ exceeds a large value) in total variation. Based on this approximation, we show that the tail event of $\int_Te^{f(t)}\,dt$ 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 $P(\int_Te^{f(t)}\,dt>b)$ with a prescribed relative accuracy.