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Efficient simulation of Brown‒Resnick processes based on variance reduction of Gaussian processes

Part of: Stochastic processes

Published online by Cambridge University Press:  29 November 2018

Marco Oesting  and
Kirstin Strokorb
Marco Oesting*
Affiliation:
University of Siegen
Kirstin Strokorb*
Affiliation:
Cardiff University
*
* Postal address: Department of Mathematics, University of Siegen, 57072 Siegen, Germany. Email address: oesting@mathematik.uni-siegen.de
** Postal address: School of Mathematics, Cardiff University, Cardiff CF24 4AG, UK. Email address: strokorbk@cardiff.ac.uk
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Abstract

Brown‒Resnick processes are max-stable processes that are associated to Gaussian processes. Their simulation is often based on the corresponding spectral representation which is not unique. We study to what extent simulation accuracy and efficiency can be improved by minimizing the maximal variance of the underlying Gaussian process. Such a minimization is a difficult mathematical problem that also depends on the geometry of the simulation domain. We extend Matheron's (1974) seminal contribution in two directions: (i) making his description of a minimal maximal variance explicit for convex variograms on symmetric domains, and (ii) proving that the same strategy also reduces the maximal variance for a huge class of nonconvex variograms representable through a Bernstein function. A simulation study confirms that our noncostly modification can lead to substantial improvements among Gaussian representations. We also compare it with three other established algorithms.

Keywords

Brown‒Resnick processGaussian processmax-stable processsimulationspatial extremesvariance reductionvariogram

MSC classification

Primary: 60G70: Extreme value theory; extremal processes 60G15: Gaussian processes
Secondary: 60G60: Random fields
Type
Original Article
Information
Advances in Applied Probability , Volume 50 , Issue 4 , December 2018 , pp. 1155 - 1175
Copyright
Copyright © Applied Probability Trust 2018 

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