A New Mixing Condition for Stationary Gaussian Processes

Abstract

A new mixing condition is proposed for the study of stationary Gaussian processes on $R^1$. If the covariance function of the process is $r$, assume $$\text{Lebesgue measure}\big\{t\mid 0 \leqslant t \leqslant T; r(t) > \frac{f(t)}{\ln t}\big\} = o(T^\beta)$$ as $T \rightarrow \infty$, for some $0 \leqslant \beta < 1$ and some $f(t) = o(1)$. The stated condition is weaker than those in common use, and yet it is shown to imply the same limit theorems on the distribution of the maximum of the process. Examples are given of processes which satisfy the new condition and not the previous ones.