Tail measures and regular variation

Bladt, Martin; Hashorva, Enkelejd; Shevchenko, Georgiy

doi:10.1214/22-ejp788

Cited by 6 publications

(2 citation statements)

References 57 publications

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“…Note in passing that B h Y can be substituted by Y in the right-hand side of (4.4) if F is shift-invariant. The identity (4.4) is shown in [8]. For the discrete setup it is shown initially in [13,23] and for case d " 1 in [22].…”

Section: Approximation Of B δmentioning

confidence: 99%

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On Berman functions

Dȩbicki¹,

Hashorva²,

Michna³

2022

Preprint

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Let Zptq " exp ´?2B H ptq ´|t| 2H ¯, t P R with B H ptq, t P R a standard fractional Brownian motion (fBm) with Hurst parameter H P p0, 1s and define for x nonnegative the Berman functionwhere the random variable R independent of Z has survival function 1{x, x ě 1 andIn this paper we consider a general random field (rf) Z that is a spectral rf of some stationary max-stable rf X and derive the properties of the corresponding Berman functions.In particular, we show that Berman functions can be approximated by the corresponding discrete ones and derive interesting representations of those functions which are of interest for Monte Carlo simulations, which are presented in this article.

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Section: Approximation Of B δmentioning

confidence: 99%

“…Motivated by the above definition, in this contribution we shall introduce the Berman functions for given δ ě 0 with respect to some non-negative rf Zptq, t P R d , d ě 1 with càdlàg sample paths (see e.g., [7,8] for the definition and properties of generalised càdlàg functions) such that…”

Section: Introductionmentioning

confidence: 99%

On Berman functions

Dȩbicki¹,

Hashorva²,

Michna³

2022

Preprint

View full text Add to dashboard Cite

show abstract

Regular Variation of Measures

Resnick

2024

The Art of Finding Hidden Risks

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Tail processes and tail measures: An approach via Palm calculus

Last

2023

Extremes

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Using an intrinsic approach, we study some properties of random fields which appear as tail fields of regularly varying stationary random fields. The index set is allowed to be a general locally compact Hausdorff Abelian group $${\mathbb G}$$ G . The values are taken in a measurable cone, equipped with a pseudo norm. We first discuss some Palm formulas for the exceedance random measure $$\xi$$ ξ associated with a stationary (measurable) random field $$Y=(Y_s)_{s\in {\mathbb G}}$$ Y = ( Y s ) s ∈ G . It is important to allow the underlying stationary measure to be $$\sigma$$ σ -finite. Then we proceed to a random field (defined on a probability space) which is spectrally decomposable, in a sense which is motivated by extreme value theory. We characterize mass-stationarity of the exceedance random measure in terms of a suitable version of the classical Mecke equation. We also show that the associated stationary measure is homogeneous, that is a tail measure. We then proceed with establishing and studying the spectral representation of stationary tail measures and with characterizing a moving shift representation. Finally we discuss anchoring maps and the candidate extremal index.

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Tail measures and regular variation

Cited by 6 publications

References 57 publications

On Berman functions

On Berman functions

Regular Variation of Measures

Tail processes and tail measures: An approach via Palm calculus

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