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 Lt 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 (Lt -ELt)/(Var(Lt))1/2 has a limiting standard normal distribution for t → ∞. The allowable rate of increase of u(t) with t is specified.
Covariance Berman Simeon
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