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

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 $c\left( d\right) =c(\{t_{1},\ldots,t_{d}\})$ function evaluations to sample $X_{n}$ at $d$ locations $t_{1},\ldots ,t_{d}\in T$. We provide an algorithm which, for any $\epsilon >0$, samples $M(t_{1}),\ldots ,M(t_{d})$ 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 H\"{o}lder continuity exponent of the process $X_{n}$ (so, if $X_{n}$ is Brownian motion, $\alpha =1/2$).