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On the probability of conjunctions of stationary Gaussian processes

Authors
Journal
Statistics & Probability Letters
0167-7152
Publisher
Elsevier
Volume
88
Identifiers
DOI: 10.1016/j.spl.2014.02.004
Keywords
  • Stationary Gaussian Processes
  • Order Statistics Processes
  • Conjunction
  • Extremes
  • Berman Sojourn Limit Theorem
  • Generalized Pickands Constant
Disciplines
  • Communication
  • Mathematics

Abstract

Abstract Let {Xi(t),t≥0},1≤i≤n be independent centered stationary Gaussian processes with unit variance and almost surely continuous sample paths. For given positive constants u,T, define the set of conjunctions C[0,T],u≔{t∈[0,T]:min1≤i≤nXi(t)≥u}. Motivated by some applications in brain mapping and digital communication systems, we obtain exact asymptotic expansion of P{C[0,T],u≠ϕ}, as u→∞. Moreover, we establish the Berman sojourn limit theorem for the random process {min1≤i≤nXi(t),t≥0} and derive the tail asymptotics of the supremum of each order statistics process.

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