, Volume 24, Issue 2, pp 229–248 | Cite as

Extremes of order statistics of stationary processes

  • Krzysztof Dȩbicki
  • Enkelejd Hashorva
  • Lanpeng Ji
  • Chengxiu Ling
Original Paper


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.


Conjunction Order statistics process Albin’s conditions Generalized Albin constant Skew-Gaussian process Gumbel limit theorem 

Mathematics Subject Classification

60G10 60G70 



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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Copyright information

© Sociedad de Estadística e Investigación Operativa 2014

Authors and Affiliations

  • Krzysztof Dȩbicki
    • 1
  • Enkelejd Hashorva
    • 2
  • Lanpeng Ji
    • 2
  • Chengxiu Ling
    • 2
  1. 1.Mathematical InstituteUniversity of WrocławWrocławPoland
  2. 2.Department of Actuarial ScienceUniversity of LausanneLausanneSwitzerland

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