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Institut for Matematik

Extreme value theory for spatial random fields - with application to a Lévy-driven field

Publikation: Bidrag til tidsskrift/Konferencebidrag i tidsskrift /Bidrag til avis › Tidsskriftartikel › Forskning › peer review

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Extreme value theory for spatial random fields - with application to a Lévy-driven field. / Stehr, Mads; Rønn-Nielsen, Anders.

I: Extremes, Bind 24, 12.2021, s. 753–795 .

Publikation: Bidrag til tidsskrift/Konferencebidrag i tidsskrift /Bidrag til avis › Tidsskriftartikel › Forskning › peer review

Vancouver

Stehr M, Rønn-Nielsen A. Extreme value theory for spatial random fields - with application to a Lévy-driven field. Extremes. 2021 dec.;24:753–795 . doi: 10.1007/s10687-021-00415-5

Author

Stehr, Mads ; Rønn-Nielsen, Anders. / Extreme value theory for spatial random fields - with application to a Lévy-driven field. I: Extremes. 2021 ; Bind 24. s. 753–795 .

Bibtex

@article{781d5f7719c9441d8dc073fec2962dc1,

title = "Extreme value theory for spatial random fields - with application to a L{\'e}vy-driven field",

abstract = "First, we consider a stationary random field indexed by an increasing sequence of subsets of $\mathbb{Z}^d$. Under certain mixing and anti--clustering conditions combined with a very broad assumption on how the sequence of spatial index sets increases, we obtain an extremal result that relates a normalized version of the distribution of the maximum of the field over the index sets to the tail distribution of the individual variables. Furthermore, we identify the limiting distribution as an extreme value distribution.Secondly, we consider a continuous, infinitely divisible random field indexed by $\mathbb{R}^d$ given as an integral of a kernel function with respect to a L\'evy basis with convolution equivalent L\'evy measure. When observing the supremum of this field over an increasing sequence of (continuous) index sets, we obtain an extreme value theorem for the distribution of this supremum. The proof relies on discretization and a conditional version of the technique applied in the first part of the paper, as we condition on the high activity and light--tailed part of the field.",

keywords = "Extreme value theory, spatial models, L\'evy--based modeling, geometric probability, intrinsic volumes, convolution equivalence, random fields, Extreme value theory, spatial models, L\'evy--based modeling, geometric probability, intrinsic volumes, convolution equivalence, random fields",

author = "Mads Stehr and Anders R{\o}nn-Nielsen",

year = "2021",

month = dec,

doi = "10.1007/s10687-021-00415-5",

language = "Dansk",

volume = "24",

pages = "753–795 ",

journal = "Extremes",

issn = "1386-1999",

publisher = "Springer",

}

RIS

TY - JOUR

T1 - Extreme value theory for spatial random fields - with application to a Lévy-driven field

AU - Stehr, Mads

AU - Rønn-Nielsen, Anders

PY - 2021/12

Y1 - 2021/12

N2 - First, we consider a stationary random field indexed by an increasing sequence of subsets of $\mathbb{Z}^d$. Under certain mixing and anti--clustering conditions combined with a very broad assumption on how the sequence of spatial index sets increases, we obtain an extremal result that relates a normalized version of the distribution of the maximum of the field over the index sets to the tail distribution of the individual variables. Furthermore, we identify the limiting distribution as an extreme value distribution.Secondly, we consider a continuous, infinitely divisible random field indexed by $\mathbb{R}^d$ given as an integral of a kernel function with respect to a L\'evy basis with convolution equivalent L\'evy measure. When observing the supremum of this field over an increasing sequence of (continuous) index sets, we obtain an extreme value theorem for the distribution of this supremum. The proof relies on discretization and a conditional version of the technique applied in the first part of the paper, as we condition on the high activity and light--tailed part of the field.

AB - First, we consider a stationary random field indexed by an increasing sequence of subsets of $\mathbb{Z}^d$. Under certain mixing and anti--clustering conditions combined with a very broad assumption on how the sequence of spatial index sets increases, we obtain an extremal result that relates a normalized version of the distribution of the maximum of the field over the index sets to the tail distribution of the individual variables. Furthermore, we identify the limiting distribution as an extreme value distribution.Secondly, we consider a continuous, infinitely divisible random field indexed by $\mathbb{R}^d$ given as an integral of a kernel function with respect to a L\'evy basis with convolution equivalent L\'evy measure. When observing the supremum of this field over an increasing sequence of (continuous) index sets, we obtain an extreme value theorem for the distribution of this supremum. The proof relies on discretization and a conditional version of the technique applied in the first part of the paper, as we condition on the high activity and light--tailed part of the field.

KW - Extreme value theory

KW - spatial models

KW - L\'evy--based modeling

KW - geometric probability

KW - intrinsic volumes

KW - convolution equivalence

KW - random fields

KW - Extreme value theory

KW - spatial models

KW - L\'evy--based modeling

KW - geometric probability

KW - intrinsic volumes

KW - convolution equivalence

KW - random fields

U2 - 10.1007/s10687-021-00415-5

DO - 10.1007/s10687-021-00415-5

M3 - Tidsskriftartikel

VL - 24

SP - 753

EP - 795

JO - Extremes

JF - Extremes

SN - 1386-1999

ER -