The Annals of Mathematical Statistics

An Asymptotic 0-1 Behavior of Gaussian Processes

Clifford Qualls and Hisao Watanabe

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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.

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Ann. Math. Statist., Volume 42, Number 6 (1971), 2029-2035.

First available in Project Euclid: 27 April 2007

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Qualls, Clifford; Watanabe, Hisao. An Asymptotic 0-1 Behavior of Gaussian Processes. Ann. Math. Statist. 42 (1971), no. 6, 2029--2035. doi:10.1214/aoms/1177693070.

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