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February, 1973 A Comparison of Continuity Conditions for Gaussian Processes
M. B. Marcus
Ann. Probab. 1(1): 123-130 (February, 1973). DOI: 10.1214/aop/1176997028


Three sufficient conditions for continuity of real-valued, separable, Gaussian processes on $\mathbb{R}^1$ are compared. They are: (1) Fernique's (1964) integral condition, (2) the Kahane (1960)-Nisio (1969) condition on the spectrum of stationary processes and (3) Dudley's (1967) condition involving $\varepsilon$-entropy. Let $S_1 \equiv$ set of stationary, separable, Gaussian processes that can be proven continuous by condition $i = 1, 2, 3$. Dudley (1967) has shown that $S_1 \subseteq S_3$. It is shown here that $S_2 \subset S_1 \subset S_3$, that is, the inclusion is strict. These results extend to non-stationary processes where appropriate. The Kahane-Nisio condition is strengthened and the best possible integral condition for continuity involving the spectrum is given. A result on the $\varepsilon$-entropy of blocks in a separable Hilbert space is also of independent interest.


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M. B. Marcus. "A Comparison of Continuity Conditions for Gaussian Processes." Ann. Probab. 1 (1) 123 - 130, February, 1973.


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

zbMATH: 0265.60039
MathSciNet: MR346885
Digital Object Identifier: 10.1214/aop/1176997028

Primary: 60G15
Secondary: 46C05 , 60G10 , 60G17

Keywords: $\varepsilon$-entropy , Continuity of Gaussian processes

Rights: Copyright © 1973 Institute of Mathematical Statistics


Vol.1 • No. 1 • February, 1973
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