The Annals of Probability

Invariant transports of stationary random measures and mass-stationarity

Günter Last and Hermann Thorisson

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Abstract

We introduce and study invariant (weighted) transport-kernels balancing stationary random measures on a locally compact Abelian group. The first main result is an associated fundamental invariance property of Palm measures, derived from a generalization of Neveu’s exchange formula. The second main result is a simple sufficient and necessary criterion for the existence of balancing invariant transport-kernels. We then introduce (in a nonstationary setting) the concept of mass-stationarity with respect to a random measure, formalizing the intuitive idea that the origin is a typical location in the mass. The third main result of the paper is that a measure is a Palm measure if and only if it is mass-stationary.

Article information

Source
Ann. Probab., Volume 37, Number 2 (2009), 790-813.

Dates
First available in Project Euclid: 30 April 2009

Permanent link to this document
https://projecteuclid.org/euclid.aop/1241099929

Digital Object Identifier
doi:10.1214/08-AOP420

Mathematical Reviews number (MathSciNet)
MR2510024

Zentralblatt MATH identifier
1176.60036

Subjects
Primary: 60G57: Random measures 60G55: Point processes
Secondary: 60G60: Random fields

Keywords
Stationary random measure invariant transport-kernel allocation rule Palm measure Abelian group mass-stationarity

Citation

Last, Günter; Thorisson, Hermann. Invariant transports of stationary random measures and mass-stationarity. Ann. Probab. 37 (2009), no. 2, 790--813. doi:10.1214/08-AOP420. https://projecteuclid.org/euclid.aop/1241099929


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