Open Access
February, 1973 Sample Functions of the Gaussian Process
R. M. Dudley
Ann. Probab. 1(1): 66-103 (February, 1973). DOI: 10.1214/aop/1176997026


This is a survey on sample function properties of Gaussian processes with the main emphasis on boundedness and continuity, including Holder conditions locally and globally. Many other sample function properties are briefly treated. The main new results continue the program of reducing general Gaussian processes to "the" standard isonormal linear process $L$ on a Hilbert space $H$, then applying metric entropy methods. In this paper Holder conditions, optimal up to multiplicative constants, are found for wide classes of Gaussian processes. If $H$ is $L^2$ of Lebesgue measure on $R^k, L$ is called "white noise." It is proved that we can write $L = P(D)\lbrack x\rbrack$ in the distribution sense where $x$ has continuous sample functions if $P(D)$ is an elliptic operator of degree $> k/2$. Also $L$ has continuous sample functions when restricted to indicator functions of sets whose boundaries are more than $k - 1$ times differentiable in a suitable sense. Another new result is that for the Levy(-Baxter) theorem $\int^1_0(dx_t)^2 = 1$ on Brownian motion, almost sure convergence holds for any sequence of partitions of mesh $o(1/\log n)$. If partitions into measurable sets other than intervals are allowed, the above is best possible: $\mathscr{O}(1/\log n)$ is insufficient.


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R. M. Dudley. "Sample Functions of the Gaussian Process." Ann. Probab. 1 (1) 66 - 103, February, 1973.


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

zbMATH: 0261.60033
MathSciNet: MR346884
Digital Object Identifier: 10.1214/aop/1176997026

Primary: 60G15
Secondary: 60G17 , 60G20

Keywords: Gaussian processes , Gaussian processes , generalized stochastic processes , Holder conditions , Metric entropy , sample functions , Sample path properties , White noise

Rights: Copyright © 1973 Institute of Mathematical Statistics

Vol.1 • No. 1 • February, 1973
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