The Annals of Probability

Sample Functions of the Gaussian Process

R. M. Dudley

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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.

Article information

Ann. Probab., Volume 1, Number 1 (1973), 66-103.

First available in Project Euclid: 19 April 2007

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 60G15: Gaussian processes
Secondary: 60G17: Sample path properties 60G20: Generalized stochastic processes

Gaussian processes sample path properties generalized stochastic processes Gaussian processes sample functions white noise Holder conditions metric entropy


Dudley, R. M. Sample Functions of the Gaussian Process. Ann. Probab. 1 (1973), no. 1, 66--103. doi:10.1214/aop/1176997026.

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