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December, 1973 Continuity of Gaussian Processes and Random Fourier Series
M. B. Marcus
Ann. Probab. 1(6): 968-981 (December, 1973). DOI: 10.1214/aop/1176996804


This paper is mainly a survey of results on the problem of finding necessary and sufficient conditions for a Gaussian process to be continuous. The relationship between this problem and the same one for random Fourier series is explored. Some new results are presented that give continuity conditions for stationary Gaussian processes in terms of the spectrum of the process. Let $X(t)$ be a real-valued stationary Gaussian process; $EX(t) = 0, EX^2(t) = 1$. Define $F$ by the equation $EX(t + h)X(t) = \int^\infty \cos \lambda h dF(\lambda)$. Assume that $F(\lambda)$ is concave for $\lambda \geqq\lambda_0 > 0$ then $X(t)$ is continuous a.s. if and only if $$\int^\infty \frac{(1 - F(x))^{\frac{1}{2}}}{x(\log x)^{\frac{1}{2}}} dx < \infty.$$ A similar result holds for Fourier series with normal coefficients.


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M. B. Marcus. "Continuity of Gaussian Processes and Random Fourier Series." Ann. Probab. 1 (6) 968 - 981, December, 1973.


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

zbMATH: 0277.60022
MathSciNet: MR356202
Digital Object Identifier: 10.1214/aop/1176996804

Primary: 60G15
Secondary: 60G17 , 60G20

Keywords: Gaussian processes , random Fourier series , sample functions

Rights: Copyright © 1973 Institute of Mathematical Statistics

Vol.1 • No. 6 • December, 1973
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