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Efficient Monte Carlo for high excursions of Gaussian random fields. (English) Zbl 1251.60031

The authors of this paper deal with Monte Carlo methods and continuous Gaussian random fields of the form \(f: T\times \Omega\rightarrow \mathbb{R}\) over a \(d\)-dimensional compact set \(T\subset \mathbb{R}^d\). To be more precise, the paper deals with the design and analysis of Monte Carlo methods for computing tail probabilities of the form \[ w(b)=\text{P}(\max_{t\in T} f(t)>b), \] and conditional expectations \[ \text{E}(\Gamma(f)|\max_{t\in T} f(t)>b), \] as \(b\rightarrow \infty\), where \(\Gamma\) is a positive and bounded functional of the field. Estimating these probabilities and conditional expectations accurately (in relative terms) by the use of an approach using naive Monte Carlo would require a computational cost that is exponential in \(b\). However, in this paper, the authors introduce simulation estimators that work for a general class of Gaussian fields (assuming that the mean and covariance functions are Hölder continuous) and require at most a polynomial number of function evalutations in \(b\). Under the assumption of additional smoothness, this result can be further improved to estimators that are strongly efficient, by which is meant that their associated coefficient of variation is uniformly bounded as \(b\) tends to infinity.

MSC:

60G15 Gaussian processes
65C05 Monte Carlo methods
60G60 Random fields
62G32 Statistics of extreme values; tail inference
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References:

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