An Asymptotic 0-1 Behavior of Gaussian Processes

Abstract

Let $\{X(t), -\infty < t < \infty\}$ be a stationary Gaussian process with covariance function satisfying: (1) $r(t) = 1 - C|t|^\alpha + o(|t|^\alpha)$ as $t \rightarrow 0: C > 0, 0 < \alpha \leqq 2$; and (2) $r(t) = O(t^{-\gamma})$ as $t \rightarrow \infty: \gamma > 0$. Then for all positive increasing functions $\phi(t)$ on $\lbrack a, \infty), P\lbrack X(t) > \phi(t)$ infinitely often $\rbrack = 0$ or 1 as $\int^\infty_a \phi(t)^{2/\alpha-1} \exp\{-\phi^2(t)/2\} dt < \infty$ or $= \infty$. This result generalizes the paper of Watanabe [Trans. Amer. Math. Soc. 148 233-248] by replacing his condition that $r(t) = o(1/t)$ as $t \rightarrow \infty$ by condition (2). Our result is extended also to the nonstationary process treated by Watanabe. Our proof treats the problem as a crossing problem using a recent result of Pickands [Trans. Amer. Math. Soc. 145 51-73] and a modification of the Borel lemmas.