## The Annals of Probability

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

### Levy Random Measures

#### 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

**Permanent link to this document**

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**

links.jstor.org

**Subjects**

Primary: 60G55: Point processes

Secondary: 60G30: Continuity and singularity of induced measures 60H99: None of the above, but in this section 60J99: None of the above, but in this section 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

**Keywords**

Levy random measure Markov property conditional independence submeasure multiplicative functional Gibbs state Gibbs random measure Markov random field Poisson random measure

#### Citation

Karr, Alan F. Levy Random Measures. Ann. Probab. 6 (1978), no. 1, 57--71. doi:10.1214/aop/1176995610. https://projecteuclid.org/euclid.aop/1176995610

#### Corrections

- See Correction: Alan F. Karr. Note: Correction to "Levy Random Measures". Ann. Probab., Volume 7, Number 6 (1979), 1098--1098.Project Euclid: euclid.aop/1176994908