Vol. 36, No. 1, 1971

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Excursions above high levels for stationary Gaussian processes

Simeon M. Berman

Vol. 36 (1971), No. 1, 63–79
Abstract

Let X(t), t 0, be a real valued stationary Gaussian process with mean 0, variance 1, covariance function r(t), and continuous sample functions. For u > 0 and T > Olet L be the Lebesgue measure of the set {t : 0 t T,X(t) > u}, i.e., the time spent above u in [0,T]. This paper proves: If r is nonperiodic, and r(t) = 1 12γ2t2 + o(t2),t 0, for some γ > 0, then the conditional distribution of γuL, given L > 0, converges for u →∞ to the distribution 1 exp(x28).

Mathematical Subject Classification
Primary: 60.50
Milestones
Received: 3 February 1970
Published: 1 January 1971
Authors
Simeon M. Berman