A Central Limit Theorem for Extreme Sojourn Times of Stationary Gaussian Processes

  • Simeon M. Berman
Conference paper
Part of the Lecture Notes in Statistics book series (LNS, volume 51)


Let X(t), t ≥ 0, be a real measurable stationary Gaussian process with mean 0 and covariance function r(t). For a given measurable function u(t)such that u(t)→ ∞, for t → ∞, let L t be the sojourn time of X(s), 0 ≤ s ≤ t, above u(t). Assume that the spectral distribution function in the representation of r(t) is absolutely continuous; then r(t) also has the representation r(t) = ∫ b(t + s)b(s)ds, where b ∈ L2. The main result is: If b ∈ L1, and if u(t)increases sufficiently slowly, then (L t -EL t )/(Var(L t ))1/2 has a limiting standard normal distribution for t → ∞. The allowable rate of increase of u(t) with t is specified.


Central Limit Theorem Gaussian Process Standard Normal Distribution Sojourn Time Small Order 
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Copyright information

© Springer-Verlag Berlin Heidelberg 1989

Authors and Affiliations

  • Simeon M. Berman
    • 1
  1. 1.Courant Institute of Mathematical SciencesNew York UniversityNew YorkUSA

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