Ergodic decompositions of stationary max-stable processes in terms of their spectral functions

Dombry, Clément; Kabluchko, Zakhar

doi:10.1016/j.spa.2016.10.001

Cited by 32 publications

(47 citation statements)

References 27 publications

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“…This representation is related to the mixed moving maximum representation of max-stable process see e.g. Dombry and Kabluchko (2017) and Section 2.5.…”

Section: Stochastic Representation Of Tail Measuresmentioning

confidence: 99%

Tail measure and spectral tail process of regularly varying time series

Dombry¹,

Hashorva²,

Soulier³

2018

Ann. Appl. Probab.

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The goal of this paper is an exhaustive investigation of the link between the tail measure of a regularly varying time series and its spectral tail process, independently introduced in Owada and Samorodnitsky (2012) and Basrak and Segers (2009). Our main result is to prove in an abstract framework that there is a one to one correspondance between these two objets, and given one of them to show that it is always possible to build a time series of which it will be the tail measure or the spectral tail process. For non negative time series, we recover results explicitly or implicitly known in the theory of maxstable processes. Tail measures on a metric space.2.1. Framework. The mathematical setting is the following. Let pE, Eq be a measurable cone, that is a measurable space together with a multiplication by positive scalars pu, xq P p0, 8qˆE Þ Ñ ux P E , which is measurable with respect to the product σ-field Bp0, 8q b E{E and satisfies 1x " x , upvpxqq " puvqx , u, v ą 0 , x P E .We assume that the cone admits a zero element 0 E P E such that u0 E " 0 E for all u ą 0 and that it is endowed with a pseudonorm, i.e. a measurable function }¨} E : E Þ Ñ r0, 8q such that }ux} E " u}x} E for all u ą 0, x P E and }x} E " 0 implies x " 0 E . The triangle inequality is not required.The space E Z of E-valued sequences is endowed with the cylinder σ-algebra F " E bZ and a generic sequence is denoted x " px h q hPZ . The sequence identically equal to 0 E is denoted by imsart-aap ver.

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“…This representation is related to the mixed moving maximum representation of max-stable process see e.g. Dombry and Kabluchko (2017) and Section 2.5.…”

Section: Stochastic Representation Of Tail Measuresmentioning

confidence: 99%

Tail measure and spectral tail process of regularly varying time series

Dombry¹,

Hashorva²,

Soulier³

2018

Ann. Appl. Probab.

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“…Throughout this section we assume that ξ W possesses a M3 representation which amounts to assuming one of the equivalent conditions below; for details see Wang and Stoev (2010) and Theorem 2 in Dombry and Kabluchko (2016). Condition 1.…”

Section: A Connection To Mixed Moving Maxima Processesmentioning

confidence: 99%

Generalized Pickands constants and stationary max-stable processes

2017

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Abstract. Pickands constants play a crucial role in the asymptotic theory of Gaussian processes. They are commonly defined as the limits of a sequence of expectations involving fractional Brownian motions and, as such, their exact value is often unknown. Recently, Dieker and Yakir (2014) derived a novel representation of Pickands constant as a simple expected value that does not involve a limit operation. In this paper we show that the notion of Pickands constants and their corresponding Dieker-Yakir representations can be extended to a large class of stochastic processes, including general Gaussian and Lévy processes. We furthermore provide a link to spatial extreme value theory and show that Pickands-type constants coincide with certain constants arising in the study of max-stable processes with mixed moving maxima representations.

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“…Each max-increment process of a max-stable random sup-measure is a stationary max-stable process. Ergodic properties of stationary max-stable processes have been recently investigated in the literature [9,15,16,33]. In particular, it is known that the max-increment processes of independently scattered random sup-measures are mixing, those of stable-regenerative random sup-measures are ergodic but not mixing, and here we show that those of Karlin random sup-measures are not ergodic.…”

Section: Introductionmentioning

confidence: 59%

A family of random sup-measures with long-range dependence

Durieu

Wang²

2018

Electron. J. Probab.

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A family of self-similar and translation-invariant random sup-measures with long-range dependence are investigated. They are shown to arise as the limit of the empirical random sup-measure of a stationary heavy-tailed process, inspired by an infinite urn scheme, where same values are repeated at several random locations. The random sup-measure reflects the long-range dependence nature of the original process, and in particular characterizes how locations of extremes appear as long-range clusters represented by random closed sets. A limit theorem for the corresponding point-process convergence is established.

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Ergodic decompositions of stationary max-stable processes in terms of their spectral functions

Cited by 32 publications

References 27 publications

Tail measure and spectral tail process of regularly varying time series

Tail measure and spectral tail process of regularly varying time series

Generalized Pickands constants and stationary max-stable processes

A family of random sup-measures with long-range dependence

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