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June, 1973 High Level Occupation Times for Continuous Gaussian Processes
Norman A. Marlow
Ann. Probab. 1(3): 388-397 (June, 1973). DOI: 10.1214/aop/1176996933


Let $\{y(\tau), 0 \leqq \tau \leqq 1\}$ be a sample continuous Gaussian process, and let $T\lbrack y, \alpha \rbrack$ denote the time that $y(\cdot)$ spends above level $\alpha:$ $$T\lbrack y, \alpha \rbrack = \int^1_0 V(y(\tau) - \alpha) d\tau,$$ where $V(x) = 0$ or 1 according as $x \leqq 0$ or $x > 0.$ In this paper it is proved that, as $\alpha \rightarrow \infty,$ $$P\{T\lbrack y, \alpha \rbrack > \beta\} = \exp \{-(\alpha^2/2)\mathbf{k}_\beta(1 + o(1))\}$$ where $k_\beta$ is a particular functional of the covariance function of the process.


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Norman A. Marlow. "High Level Occupation Times for Continuous Gaussian Processes." Ann. Probab. 1 (3) 388 - 397, June, 1973.


Published: June, 1973
First available in Project Euclid: 19 April 2007

zbMATH: 0262.60021
MathSciNet: MR362469
Digital Object Identifier: 10.1214/aop/1176996933

Primary: 60G15
Secondary: 41A60

Keywords: asymptotic distribution , Gaussian processes , high level occupation times

Rights: Copyright © 1973 Institute of Mathematical Statistics

Vol.1 • No. 3 • June, 1973
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