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Extremes

pp 1–19 | Cite as

Canonical spectral representation for exchangeable max-stable sequences

  • Jan-Frederik MaiEmail author
Article
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Abstract

The set \(\mathfrak {L}\) of infinite-dimensional, symmetric stable tail dependence functions associated with exchangeable max-stable sequences of random variables with unit Fréchet margins is shown to be a simplex. Except for a single element, the extremal boundary of \(\mathfrak {L}\) is in one-to-one correspondence with the set \(\mathfrak {F}_{1}\) of distribution functions of non-negative random variables with unit mean. Consequently, each \(\ell \in \mathfrak {L}\) is uniquely represented by a pair (b, µ) of a constant b and a probability measure µ on \(\mathfrak {F}_{1}\). A canonical stochastic construction for arbitrary exchangeable max-stable sequences and a stochastic representation for the Pickands dependence measure of finite-dimensional margins of l are immediate corollaries. As by-products, a canonical analytical description and an associated canonical Le Page series representation for non-decreasing stochastic processes that are strongly infinitely divisible with respect to time are obtained.

Keywords

Exchangeable sequence Max-stable sequence Stable tail dependence function Extreme-value copula Strong IDT process Pickands representation 

AMS 2000 Subject Classifications

60G09 60G70 60E07 

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Notes

Acknowledgments

Inspiring discussions with Paul Ressel and his helpful comments on earlier versions of this manuscript are gratefully acknowledged. His remarks in particular made me aware of the somewhat special role of the point lp. Helpful comments by the anonymous referees and the handling editor are also gratefully acknowledged.

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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Technische Universität MünchenGarching-HochbrückGermany

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