Open Access
February, 1978 Levy Random Measures
Alan F. Karr
Ann. Probab. 6(1): 57-71 (February, 1978). DOI: 10.1214/aop/1176995610


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 $\mathbf{Z}^\nu$.


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Alan F. Karr. "Levy Random Measures." Ann. Probab. 6 (1) 57 - 71, February, 1978.


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

zbMATH: 0374.60063
MathSciNet: MR455115
Digital Object Identifier: 10.1214/aop/1176995610

Primary: 60G55
Secondary: 60G30 , 60H99 , 60J99 , 60K35

Keywords: Conditional independence , Gibbs random measure , Gibbs state , Levy random measure , Markov property , Markov random field , Multiplicative functional , Poisson random measure , submeasure

Rights: Copyright © 1978 Institute of Mathematical Statistics

Vol.6 • No. 1 • February, 1978
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