Extremes of vector-valued Gaussian processes |
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| Affiliation: | 1. Center for Informatics and Systems, University of Coimbra, Coimbra, Portugal;2. Computer Vision Center, Autonomous University of Barcelona, Barcelona, Spain;1. CES, University Paris 1 Pantheon-Sorbonne, France;2. The Institute for Information Transmission Problems, Bolshoy Karetny 19, 127051 Moscow, Russia;3. University of Bern, Institute of Mathematical Statistics and Actuarial Science, Alpeneggstrasse 22, 3012 Bern, Switzerland |
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| Abstract: | The seminal papers of Pickands (Pickands, 1967; Pickands, 1969) paved the way for a systematic study of high exceedance probabilities of both stationary and non-stationary Gaussian processes. Yet, in the vector-valued setting, due to the lack of key tools including Slepian’s Lemma, there has not been any methodological development in the literature for the study of extremes of vector-valued Gaussian processes. In this contribution we develop the uniform double-sum method for the vector-valued setting, obtaining the exact asymptotics of the high exceedance probabilities for both stationary and n on-stationary Gaussian processes. We apply our findings to the operator fractional Brownian motion and Ornstein–Uhlenbeck process. |
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| Keywords: | High exceedance probability Vector-valued Gaussian process Operator fractional Ornstein–Uhlenbeck processes Operator fractional Brownian motion Uniform double-sum method Vector-valued Borell-TIS inequality |
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