Invariant transports of stationary random measures and mass-stationarity

Invariant transports of stationary random measures and mass-stationarity

Günter Last and Hermann Thorisson

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

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.

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Primary Subjects: 60G57, 60G55

Secondary Subjects: 60G60

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

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Permanent link to this document: http://projecteuclid.org/euclid.aop/1241099929
Digital Object Identifier: doi:10.1214/08-AOP420
Zentralblatt MATH identifier: 05558300
Mathematical Reviews number (MathSciNet): MR2510024

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