On Optimal Exact Simulation of Max-Stable and Related Random Fields

Abstract

We consider the random field M(t)=\sup_{n\geq 1}\big\{-\log A_{n}+X_{n}(t)\big\}\,,\qquad t\in T\, for a set $T\subset {\mathbb{R}}^m$, where $(X_n)$ is an iid 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 $c\left(d\right)=c(\{t_1,\dots ,t_d\})$ function evaluations to sample $X_n$ at $d$ locations $t_1,\dots ,t_d\in T$. We provide an algorithm which, for any $\epsilon >0$, samples $M(t_1),\dots ,M(t_d)$ with complexity $o(c(d)d^{\epsilon })$. 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, $\alpha =1/2$).

References in corpus (3)

• Exact simulation of max-stable processes
• Sampling Point Processes on Stable Unbounded Regions and Exam Simulation of Queues
• Perfect Sampling of Generalized Jackson Network

Cited by in corpus (4)

• $\epsilon$-strong simulation of the convex minorants of stable processes and meanders
• Conditionally Max-stable Random Fields based on log Gaussian Cox Processes
• Malliavin-based Multilevel Monte Carlo Estimators for Densities of Max-stable Processes
• A Dichotomy for Sampling Barrier-Crossing Events of Random Walks with Regularly Varying Tails