Characterization and construction of max-stable processes

by Kirstin Strokorb
Doctoral thesis
Date of Examination:2013-07-02
Date of issue:2013-08-30
Advisor:Prof. Dr. Andrea Krajina
Referee:Prof. Dr. Andrea Krajina
Referee:Prof. Dr. Robert Schaback
Referee:Prof. Dr. Martin Schlather
Referee:Prof. Dr. Ilya Molchanov
Persistent Address: http://dx.doi.org/10.53846/goediss-4022

 

 

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Abstract

English

Max-stable processes provide a natural framework to model spatial extremal scenarios. Appropriate summary statistics include the extremal coefficients and the (upper) tail dependence coefficients. In this thesis, the full set of extremal coefficients of a max-stable process is captured in the so-called extremal coefficient function (ECF) and the full set of upper tail dependence coefficients in the tail correlation function (TCF). Chapter 2 deals with a complete characterization of the ECF in terms of negative definiteness. For each ECF a corresponding max-stable process is constructed, which takes an exceptional role among max-stable processes with identical ECF. This leads to sharp lower bounds for the finite dimensional distributions of arbitrary max-stable processes in terms of its ECF. Chapters 3 and 4 are concerned with the class of TCFs. Chapter 3 exhibits this class as an infinite-dimensional compact convex polytope. It is shown that the set of all TCFs (of not necessarily max-stable processes) coincides with the set of TCFs stemming from max-stable processes. Chapter 4 compares the TCFs of widely used stationary max-stable processes such as Mixed Moving Maxima, Extremal Gaussian and Brown-Resnick processes. Finally, in Chapter 5, Brown-Resnick processes on the sphere and other spaces admitting a compact group action are considered and a Mixed Moving Maxima representation is derived.
Keywords: extreme value theory; max-stable process; extremal coefficient; negative definite; tail correlation; positive definite; dependency set; harmonic analysis; mixed moving maxima; Brown-Resnick process; extremal Gaussian process; tail dependence
 
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