A Note on a O-1 Law for Stationary Gaussian Processes,

Abstract : It is shown that a stationary Gaussian process X(t), defined on t = 1,2, attains values which exceed any given non-decreasing function f(t) infinitely often with probability zero or one. The only assumption made is that the covariance function r(t) go to zero as t goes to infinity. When r(t) is of smaller order than t sup gamma for some gamma > 0, a test is given which distinguishes between the two cases of probability zero and one. Similar results are indicated when, instead, the index t assumes all the values in the interval (0, infinity) and the sample paths of x(t) are continuous. (Author)