Central Limit Theorem for Extreme Sojourn Times of Stationary Gaussian Processes

摘要

Let X(t), t > or = O, be a real measurable stationary Gaussian process with mean O and covariance function r(t). For a given measurable function u(t) such that u(t) approaches infinity, let Lt, be the sojourn time of X(s), O < or = 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 is an element of L sub 2. The main result is: If b is an element of L sub 1, and if u(t) increases sufficiently slowly, then (L sub t - EL sub t)/Var(L sub t)1/2 has a limiting standard normal distribution for t approaches infinity. The allowable rate of increase of u(t) with t is specified. Reprints. (jhd)