## The Annals of Mathematical Statistics

### Continuity Properties of Some Gaussian Processes

Christopher Preston

#### Abstract

Let $(S, d)$ be a compact metric space; let $(\Omega, \mathscr{F}, P)$ be a probability space, and for each $t \in S$ let $X_t: \Omega \rightarrow \mathbb{R}$ be a random variable, with $E(X_t) = 0$ and such that $\{X_t\}_{t\in S}$ forms a Gaussian process. In this paper we find sufficient conditions for the Gaussian process $\{X_t\}_{t\in S}$ to admit a separable and measurable model whose sample functions are continuous with probability one. The conditions involve the covariance, $E(X_s, X_t)$, of the process and also the $\varepsilon$-entropy of $S$.

#### Article information

Source
Ann. Math. Statist., Volume 43, Number 1 (1972), 285-292.

Dates
First available in Project Euclid: 27 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aoms/1177692721

Digital Object Identifier
doi:10.1214/aoms/1177692721

Mathematical Reviews number (MathSciNet)
MR307316

Zentralblatt MATH identifier
0268.60044

JSTOR
links.jstor.org

#### Citation

Preston, Christopher. Continuity Properties of Some Gaussian Processes. Ann. Math. Statist. 43 (1972), no. 1, 285--292. doi:10.1214/aoms/1177692721. https://projecteuclid.org/euclid.aoms/1177692721