Excursions above high levels for stationary Gaussian processes

Simeon M. Berman

Vol. 36 (1971), No. 1, 63–79

DOI: 10.2140/pjm.1971.36.63

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 − 1∕2γ²t² + 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(−x²∕8).

Mathematical Subject Classification

Primary: 60.50

Milestones

Received: 3 February 1970

Published: 1 January 1971