## The Annals of Probability

### On the Increments of Multidimensional Random Fields

#### Abstract

For a nondifferentiable random field $\{X_t: t \in \mathbb{R}^N\}$ with values in $\mathbb{R}^d$, it is often easy to check that with probability 1 $\lim \inf_{s\rightarrow t}\|X_s - X_t\|/\sigma(s, t) = 0$ and $\lim \sup_{s\rightarrow t}\|X_s - X_t\|/\sigma(s, t) = \infty$ for a.e. $t$, where $\sigma^2(s, t) = E\|X_s - X_t\|^2$. In this note we discuss the "proportion" of $s$'s near $t$ for which $\|X_s - X_t\|/\sigma(s, t)$ is small or large.

#### Article information

Source
Ann. Probab., Volume 6, Number 1 (1978), 151-158.

Dates
First available in Project Euclid: 19 April 2007

https://projecteuclid.org/euclid.aop/1176995620

Digital Object Identifier
doi:10.1214/aop/1176995620

Mathematical Reviews number (MathSciNet)
MR461638

Zentralblatt MATH identifier
0405.60055

JSTOR