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

Jingchen Liu, Gongjun Xu

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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 ΣT ef(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 ΣT ef(t) dt exceeds a large value) in total variation. Based on this approximation, we show that the tail event of ΣT ef(t) dt is asymptotically equivalent to the tail event of supT 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(ΣT ef(t) dt > b) with a prescribed relative accuracy.

Original languageEnglish (US)
Pages (from-to)1691-1738
Number of pages48
JournalAnnals of Applied Probability
Volume24
Issue number4
DOIs
StatePublished - Aug 2014

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Keywords

  • Change of measure
  • Efficient simulation
  • Gaussian process

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