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Tail processes and tail measures: An approach via Palm calculus

Last, Günter 1
1 Institut für Stochastik (STOCH), Karlsruher Institut für Technologie (KIT)


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 $\mathbb{G}$. The values are taken in a measurable cone, equipped with a pseudo norm. We first discuss some Palm formulas for the exceedance random measure $ξ$ associated with a stationary (measurable) random field $Y=(Y_s)_{s∈G}$. It is important to allow the underlying stationary measure to be $σ$-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.

Verlagsausgabe §
DOI: 10.5445/IR/1000160703
Veröffentlicht am 18.07.2023
Cover der Publikation
Zugehörige Institution(en) am KIT Institut für Stochastik (STOCH)
Publikationstyp Zeitschriftenaufsatz
Publikationsjahr 2023
Sprache Englisch
Identifikator ISSN: 1386-1999, 1572-915X
KITopen-ID: 1000160703
Erschienen in Extremes
Verlag Springer
Vorab online veröffentlicht am 27.06.2023
Schlagwörter Tail process, Exceedances, Tail measure, Spectral representation, Random measure, Palm measure, Stationarity, Mass-stationarity, Locally compact Abelian group, Anchoring map, Candidate extremal index
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