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Rare-event simulation and efficient discretization for the supremum of Gaussian random fields

Part of: Nonparametric inference Stochastic processes Probabilistic methods, simulation and stochastic differential equations

Published online by Cambridge University Press:  21 March 2016

Xiaoou Li  and
Jingchen Liu
Xiaoou Li*
Affiliation:
Columbia University
Jingchen Liu*
Affiliation:
Columbia University
*
∗ Postal address: Department of Statistics, Columbia University, 1255 Amsterdam Avenue, New York, NY 10027, USA.
∗ Postal address: Department of Statistics, Columbia University, 1255 Amsterdam Avenue, New York, NY 10027, USA.
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Abstract

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In this paper we consider a classic problem concerning the high excursion probabilities of a Gaussian random field f living on a compact set T. We develop efficient computational methods for the tail probabilities ℙ{supTf(t) > b}. For each positive ε, we present Monte Carlo algorithms that run in constant time and compute the probabilities with relative error ε for arbitrarily large b. The efficiency results are applicable to a large class of Hölder continuous Gaussian random fields. Besides computations, the change of measure and its analysis techniques have several theoretical and practical indications in the asymptotic analysis of Gaussian random fields.

Keywords

Gaussian random fieldhigh-level excursionMonte Carlotail distributionefficiency

MSC classification

Primary: 60G15: Gaussian processes 65C05: Monte Carlo methods
Secondary: 60G60: Random fields 62G32: Statistics of extreme values; tail inference
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 47 , Issue 3 , September 2015 , pp. 787 - 816
Copyright
Copyright © Applied Probability Trust 2015 

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