SPECTRAL CONDITIONS FOR SOJOURN AND EXTREME VALUE LIMIT-THEOREMS FOR GAUSSIAN-PROCESSES

Let X(t), t greater-than-or-equal-to 0, be a stationary Gaussian process, and define the sojourn time L(u)(t) = mes{s: 0 less-than-or-equal-to s less-than-or-equal-to t, X (s) > u} and the maximum Z(t) = max(X(s): 0 less-than-or-equal-to s less-than-or-equal-to t). Limit theorems for the distributions of L(u)(t) and Z(t), for t, u --> infinity, are obtained under specified conditions on the spectral density of the process. The results supplement earlier theorems obtained under suitable conditions on the covariance function.