Maxima and high level excursions of stationary Gaussian processes

Let X(t), t > 0, be a stationary Gaussian process with mean 0, variance 1 and covariance function r(t). The sample functions are assumed to be continuous on every interval. Let r(t) be continuous and nonperiodic. Suppose that there exists a, 0 0, satisfying (0.1) lim g(ct) 1, for every c > 0, t-~o g(t) such that (0.2) 1 -r(t) g(it J)Jt I, t O . For u > 0, let v be defined (in terms of u) as the unique solution of (0.3) u2g(1/v)v= 1. Let IA be the indicator of the event A; then rT JI[x(s) > u] ds represents the time spent above u by X(s), 0 u] ds, given that it is positive, converges for fixed T and u -* oo to a limiting distribution TF, which depends only on a but not on T or g. Let F(A) be the spectral distribution function corresponding to r(t). Let F(1)(A) be the iterated p-fold convolution of F(A). If, in addition to (0.2), it is assumed that (0.5) (P) is absolutely continuous for some p > 0, then max (X(s): 0<s < t), properly normalized, has, for t -* oo, the limiting extreme value distribution exp (e x). Received by the editors September 22, 1970. AMS 1970 subject classifications. Primary 60G10, 60G15, 60G17; Secondary 60F99.