## The Annals of Probability

### Sample Functions of the Gaussian Process

R. M. Dudley

#### Abstract

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

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

Dates
First available in Project Euclid: 19 April 2007

https://projecteuclid.org/euclid.aop/1176997026

Digital Object Identifier
doi:10.1214/aop/1176997026

Mathematical Reviews number (MathSciNet)
MR346884

Zentralblatt MATH identifier
0261.60033

JSTOR