## Bernoulli

• Bernoulli
• Volume 25, Number 4A (2019), 2949-2981.

### On logarithmically optimal exact simulation of max-stable and related random fields on a compact set

#### Abstract

We consider the random field \begin{equation*}M(t)=\mathop{\mathrm{sup}}_{n\geq1}\{-\log A_{n}+X_{n}(t)\},\qquad t\in T,\end{equation*} 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, $\alpha=1/2$).

#### Article information

Source
Bernoulli, Volume 25, Number 4A (2019), 2949-2981.

Dates
Revised: September 2018
First available in Project Euclid: 13 September 2019

https://projecteuclid.org/euclid.bj/1568362048

Digital Object Identifier
doi:10.3150/18-BEJ1076

Mathematical Reviews number (MathSciNet)
MR4003570

Zentralblatt MATH identifier
07110117

#### Citation

Liu, Zhipeng; Blanchet, Jose H.; Dieker, A.B.; Mikosch, Thomas. On logarithmically optimal exact simulation of max-stable and related random fields on a compact set. Bernoulli 25 (2019), no. 4A, 2949--2981. doi:10.3150/18-BEJ1076. https://projecteuclid.org/euclid.bj/1568362048

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