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On logarithmically optimal exact simulation of max-stable and related random fields on a compact set. (English) Zbl 1428.62426
Summary: We consider the random field \[M(t)=\sup_{n\geq1}\{-\log A_n+X_n(t)\},\qquad t\in T,\] for a set \(T\subset\mathbb{R}^m\), where \((X_n)\) is an i.i.d. sequence of centered Gaussian random fields on \(T\) and \(0<A_1<A_2<\cdots\) are the arrivals of a general renewal process on \((0,\infty)\), independent of \((X_n)\). In particular, a large class of max-stable random fields with Gumbel marginals have such a representation. Assume that one needs \(c(d)=c(\{t_1,\ldots,t_d\})\) function evaluations to sample \(X_n\) at \(d\) locations \(t_1,\ldots,t_d\in T\). We provide an algorithm which samples \(M(t_1),\ldots,M(t_d)\) with complexity \(O(c(d)^{1+o(1)})\) as measured in the \(L_p\) norm sense for any \(p\ge1\). Moreover, if \(X_n\) has an a.s. converging series representation, then \(M\) can be a.s. approximated with error \(\delta\) uniformly over \(T\) and with complexity \(O(1/(\delta\log(1/\delta))^{1/\alpha})\), where \(\alpha\) relates to the Hölder continuity exponent of the process \(X_n\) (so, if \(X_n\) is Brownian motion, \( \alpha=1/2)\).

60G70 Extreme value theory; extremal stochastic processes
60G60 Random fields
60-08 Computational methods for problems pertaining to probability theory
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