Extremes of vector-valued Gaussian processes: Exact asymptotics

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Type
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Publications
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Title
Extremes of vector-valued Gaussian processes: Exact asymptotics
Journal
Stochastic Processes and their Applications
Author(s)
Dȩbicki  K., Hashorva  E., Ji  L., Tabiś  K.
ISSN
0304-4149 (Print)
Publication state
Published
Issued date
11/2015
Peer-reviewed
Oui
Volume
125
Number
11
Pages
4039-4065
Language
english
Abstract
Let {X-i(t), t >= 0}, 1 <= i <= n be mutually independent centered Gaussian processes with almost surely continuous sample paths. We derive the exact asymptotics of
P (there exists(t epsilon[0,T])for all(i=1),..,X-n(i)(t) > u) as u -> infinity
for both locally stationary X-i 's and X-i 's with a non-constant generalized variance function. Additionally, we analyze properties of multidimensional counterparts of the Pickands and Piterbarg constants that appear in the derived asymptotics. Important by-products of this contribution are the vector-process extensions of the Piterbarg inequality, the Borell-TIS inequality, the Slepian lemma and the Pickands Piterbarg lemma which are the main pillars of the extremal theory of vector-valued Gaussian processes.
Keywords
Gaussian process, Conjunction, Extremes, Double-sum method, Slepian lemma, Borell-TIS inequality, Piterbarg inequality, Generalized Pickands constant, Generalized Piterbarg constant, Pickands-Piterbarg lemma
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28/05/2015 8:57
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21/08/2019 6:09
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