Open Access
August, 1974 Asymptotic Maxima of Continuous Gaussian Processes
M. B. Marcus
Ann. Probab. 2(4): 702-713 (August, 1974). DOI: 10.1214/aop/1176996613


Let $X(t)$ be a stationary Gaussian process with continuous sample paths. The behavior of $|X(t)|$ as $t \rightarrow \infty$ is considered. In particular, conditions on the spectrum of the process are given which determine whether $\lim \sup_{t\rightarrow\infty}|X(t)|/(\log t)^{\frac{1}{2}} = \operatorname{Const.} > 0$. These conditions are complete except when the spectrum of the process is continuous-singular. The main concern of this paper is to study the asymptotic behavior of some specific examples of $X(t)$ with continuous-singular spectra. Many examples are given showing the asymptotic behavior of stationary Gaussian processes with discrete spectra and their indefinite integrals.


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M. B. Marcus. "Asymptotic Maxima of Continuous Gaussian Processes." Ann. Probab. 2 (4) 702 - 713, August, 1974.


Published: August, 1974
First available in Project Euclid: 19 April 2007

zbMATH: 0304.60024
MathSciNet: MR370726
Digital Object Identifier: 10.1214/aop/1176996613

Primary: 60G15
Secondary: 60E05 , 60G17

Keywords: asymptotic rates , Maxima of Gaussian process , processes with stationary increments

Rights: Copyright © 1974 Institute of Mathematical Statistics

Vol.2 • No. 4 • August, 1974
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