TY - JOUR
T1 - On logarithmically optimal exact simulation of max-stable and related random fields on a compact set
AU - Liu, Zhipeng
AU - Blanchet, Jose H.
AU - Dieker, A.b.
AU - Mikosch, Thomas
PY - 2019
Y1 - 2019
N2 - We consider the random field M(t) = nsup ≥1 {− log An + Xn(t)}, t ∈ T, for a set T ⊂ Rm, where (Xn) is an i.i.d. sequence of centered Gaussian random fields on T and 0 < A1 < A2 < · · · are the arrivals of a general renewal process on (0, ∞), independent of (Xn). In particular, a large class of max-stable random fields with Gumbel marginals have such a representation. Assume that one needs c(d) = c({t1, . . ., td}) function evaluations to sample Xn at d locations t1, . . ., td ∈ T . We provide an algorithm which samples M(t1), . . ., M(td) with complexity O(c(d)1+o(1)) as measured in the Lp norm sense for any p ≥ 1. Moreover, if Xn has an a.s. converging series representation, then M can be a.s. approximated with error δ uniformly over T and with complexity O(1/(δ log(1/δ))1/α), where α relates to the Hölder continuity exponent of the process Xn (so, if Xn is Brownian motion, α = 1/2).
AB - We consider the random field M(t) = nsup ≥1 {− log An + Xn(t)}, t ∈ T, for a set T ⊂ Rm, where (Xn) is an i.i.d. sequence of centered Gaussian random fields on T and 0 < A1 < A2 < · · · are the arrivals of a general renewal process on (0, ∞), independent of (Xn). In particular, a large class of max-stable random fields with Gumbel marginals have such a representation. Assume that one needs c(d) = c({t1, . . ., td}) function evaluations to sample Xn at d locations t1, . . ., td ∈ T . We provide an algorithm which samples M(t1), . . ., M(td) with complexity O(c(d)1+o(1)) as measured in the Lp norm sense for any p ≥ 1. Moreover, if Xn has an a.s. converging series representation, then M can be a.s. approximated with error δ uniformly over T and with complexity O(1/(δ log(1/δ))1/α), where α relates to the Hölder continuity exponent of the process Xn (so, if Xn is Brownian motion, α = 1/2).
U2 - 10.3150/18-bej1076
DO - 10.3150/18-bej1076
M3 - Journal article
SN - 1350-7265
VL - 25
SP - 2949
EP - 2981
JO - Bernoulli
JF - Bernoulli
IS - 4A
ER -