Open Access
February, 1978 On the Increments of Multidimensional Random Fields
Donald Geman, Joel Zinn
Ann. Probab. 6(1): 151-158 (February, 1978). DOI: 10.1214/aop/1176995620


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.


Download Citation

Donald Geman. Joel Zinn. "On the Increments of Multidimensional Random Fields." Ann. Probab. 6 (1) 151 - 158, February, 1978.


Published: February, 1978
First available in Project Euclid: 19 April 2007

zbMATH: 0405.60055
MathSciNet: MR461638
Digital Object Identifier: 10.1214/aop/1176995620

Primary: 60G10
Secondary: 60G15 , 60G17

Keywords: approximate limit , Gaussian process , Random field , Stationary increments

Rights: Copyright © 1978 Institute of Mathematical Statistics

Vol.6 • No. 1 • February, 1978
Back to Top