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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)

Abstract

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.

Keywords

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

  1. 1.
    Berman, S. M., Occupation times of stationary Gaussian processes, J. Applied Probability 7 (1970) 721 - 733.MATHCrossRefGoogle Scholar
  2. 2.
    Berman, S. M., Maxima and high level excursions of stationary Gaussian processes, Trans. Amer. Math. Soc. 160 (1971) 65 - 85.MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Berman, S. M., High level sojourns for strongly dependent Gaussian processes, Z. Wahrscheinlichkeitstheorie verw. Gebiete 50 (1979) 223 - 236.MATHCrossRefGoogle Scholar
  4. 4.
    Berman, S. M., A compound Poisson limit for stationary sums, and sojourns of Gaussian processes, Ann. Probability 8 (1980) 511 - 538.MATHCrossRefGoogle Scholar
  5. 5.
    Berman, S. M., Sojourns of vector Gaussian processes inside and outside spheres, Z. Wahrcheinlichkeitstheorie verw. Gebeite. 66 (1984) 529 - 542.MATHCrossRefGoogle Scholar
  6. 6.
    Cramer, H. & Leadbetter, M. R., Stationary and Related Stochastic Processes, John Wiley, New York, 1967.MATHGoogle Scholar
  7. 7.
    Doob, J. L., Stochastic Processes, John Wiley, New York, 1954.MATHGoogle Scholar
  8. 8.
    Esseen, C. G., Fourier analysis of distribution functions, Acta Math. 77 (1944) 1 - 125.MathSciNetCrossRefGoogle Scholar
  9. 9.
    Sun, T. C., Some further results on central limit theorems for nonlinear functionals of a normal stationary process, J. Math. Mech. 14 (1965) 71 - 85.MathSciNetMATHGoogle Scholar
  10. 10.
    Volkonskii, V. A. & Rozanov, Yu. A., Some limit theorems for random functions II. Theory Probability Appl. 6 (1961) 186 - 198.MathSciNetCrossRefGoogle Scholar

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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