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Extremes of order statistics of stationary processes

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Abstract

Let \(\{X_i(t),t\ge 0\}, 1\le i\le n\) be independent copies of a stationary process \(\{X(t), t\ge 0\}\). For given positive constants \(u,T\), define the set of \(r\)th conjunctions \( C_{r,T,u}:= \{t\in [0,T]: X_{r:n}(t) > u\}\) with \(X_{r:n}(t)\) the \(r\)th largest order statistics of \(X_i(t), t\ge 0, 1\le i\le n\). In numerous applications such as brain mapping and digital communication systems, of interest is the approximation of the probability that the set of conjunctions \(C_{r,T,u}\) is not empty. Imposing the Albin’s conditions on \(X\), in this paper we obtain an exact asymptotic expansion of this probability as \(u\) tends to infinity. Furthermore, we establish the tail asymptotics of the supremum of the order statistics processes of skew-Gaussian processes and a Gumbel limit theorem for the minimum order statistics of stationary Gaussian processes.

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Acknowledgments

We are grateful to the referees and the Editor-in-Chief for their careful reading and numerous suggestions which greatly improved the paper.

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Correspondence to Chengxiu Ling.

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Partial support from the Swiss National Science Foundation Project 200021-140633/1 and Marie Curie International Research Staff Exchange Scheme Fellowship within the 7th European Community Framework Programme (Grant No. RARE-318984) is kindly acknowledged. The first author also acknowledges partial support by Narodowe Centrum Nauki Grant No. 2013/09/B/ST1/01778 (2014–2016).

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Dȩbicki, K., Hashorva, E., Ji, L. et al. Extremes of order statistics of stationary processes. TEST 24, 229–248 (2015). https://doi.org/10.1007/s11749-014-0404-4

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