An exceptional max-stable process fully parameterized by its extremal coefficients

Strokorb, Kirstin ORCID: https://orcid.org/0000-0001-8748-3014 and Schlather, Martin 2015. An exceptional max-stable process fully parameterized by its extremal coefficients. Bernoulli 21 (1) , pp. 276-302. 10.3150/13-BEJ567

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Abstract

The extremal coefficient function (ECF) of a max-stable process X on some index set T assigns to each finite subset A⊂T the effective number of independent random variables among the collection {Xt}t∈A. We introduce the class of Tawn–Molchanov processes that is in a 1:1 correspondence with the class of ECFs, thus also proving a complete characterization of the ECF in terms of negative definiteness. The corresponding Tawn–Molchanov process turns out to be exceptional among all max-stable processes sharing the same ECF in that its dependency set is maximal w.r.t. inclusion. This entails sharp lower bounds for the finite dimensional distributions of arbitrary max-stable processes in terms of its ECF. A spectral representation of the Tawn–Molchanov process and stochastic continuity are discussed. We also show how to build new valid ECFs from given ECFs by means of Bernstein functions.

Item Type:	Article
Date Type:	Published Online
Status:	Published
Schools:	Mathematics
Subjects:	Q Science > QA Mathematics
Uncontrolled Keywords:	Completely alternating; Dependency set; Extremal coefficient; Max-linear model; Max-stable process; Negative definite; Semigroup; Spectrally discrete; Tawn–Molchanov process
Publisher:	Bernoulli Society for Mathematical Statistics and Probability
ISSN:	1350-7265
Funders:	DFG (RTG1023)
Date of First Compliant Deposit:	6 January 2017
Date of Acceptance:	24 September 2013
Last Modified:	06 May 2023 04:16
URI:	https://orca.cardiff.ac.uk/id/eprint/97222

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