Excursions above high levels for stationary Gaussian processes
Simeon M. Berman · Pacific Journal of Mathematics · 1971
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 > 0 let L be the Lebesgue measure of the set {t: 0 ^ t ^ T, X(t) > u} 9 i.e., the time spent above u in [0, T], This paper proves: If r is nonperiodic, and r(t) = 1-1/2 f t 2 + o(t 2 ), t -> 0, for some γ > 0, then the conditional distribution of fwL, given L > 0, converges for u -> co to the distribution 1 -exp