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

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 Σ_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 Σ_T e^f(t) dt exceeds a large value) in total variation. Based on this approximation, we show that the tail event of Σ_T e^f(t) dt is asymptotically equivalent to the tail event of sup_T γ^(t) where γ^(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(Σ_T e^f(t) dt > b) with a prescribed relative accuracy.