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2014 When are increment-stationary random point sets stationary?
Antoine Gloria
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Electron. Commun. Probab. 19: 1-14 (2014). DOI: 10.1214/ECP.v19-3288


In a recent work, Blanc, Le Bris, and Lions defined a notion of increment-stationarity for random point sets, which allowed them to prove the existence of a thermodynamic limit for two-body potential energies on such point sets (under the additional assumption of ergodicity), and to introduce a variant of stochastic homogenization for increment-stationary coefficients. Whereas stationary random point sets are increment-stationary, it is not clear a priori under which conditions increment-stationary random point sets are stationary.In the present contribution, we give a characterization of the equivalence of both notions of stationarity based on elementary PDE theory in the probability space.This allows us to give conditions on the decay of a covariance function associated with the random point set, which ensure that increment-stationary random point sets are stationary random point sets up to a random translation with bounded second moment in dimensions $d>2$. In dimensions $d=1$ and $d=2$, we show that such sufficient conditions cannot exist.


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Antoine Gloria. "When are increment-stationary random point sets stationary?." Electron. Commun. Probab. 19 1 - 14, 2014.


Accepted: 18 May 2014; Published: 2014
First available in Project Euclid: 7 June 2016

zbMATH: 1315.60016
MathSciNet: MR3208328
Digital Object Identifier: 10.1214/ECP.v19-3288

Primary: 60D05
Secondary: 39A70 , 60H25

Keywords: Random geometry , random point sets , Stochastic homogenization , thermodynamic limit


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