## The Annals of Probability

### Levy Random Measures

Alan F. Karr

#### Abstract

A Levy random measure is characterized by a conditional independence structure analogous to the Markov property. Here we introduce Levy random measures and present their basic properties. Preservation of the Levy property under transformations of random measures (e.g., change of variable, passage to a limit) and under transformations of the probability laws of random measures is investigated. One random measure is said to be a submeasure of a second random measure if its probability law is absolutely continuous with respect to that of the second. We show that if the second measure is a Levy random measure then the submeasure is Levy if and only if the Radon-Nikodym derivative satisfies a natural factorization condition. These results are applied to extend the theories of Gibbs states on bounded sets in $\mathbb{R}^\nu$ and $\mathbf{Z}^\nu$.

#### Article information

Source
Ann. Probab., Volume 6, Number 1 (1978), 57-71.

Dates
First available in Project Euclid: 19 April 2007

https://projecteuclid.org/euclid.aop/1176995610

Digital Object Identifier
doi:10.1214/aop/1176995610

Mathematical Reviews number (MathSciNet)
MR455115

Zentralblatt MATH identifier
0374.60063

JSTOR