## The Annals of Probability

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

### 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

**Permanent link to this document**

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**

links.jstor.org

**Subjects**

Primary: 60G10: Stationary processes

Secondary: 60G15: Gaussian processes 60G17: Sample path properties

**Keywords**

Random field approximate limit Gaussian process stationary increments

#### Citation

Geman, Donald; Zinn, Joel. On the Increments of Multidimensional Random Fields. Ann. Probab. 6 (1978), no. 1, 151--158. doi:10.1214/aop/1176995620. https://projecteuclid.org/euclid.aop/1176995620