Open Access
March 2010 Ergodic theory, Abelian groups and point processes induced by stable random fields
Parthanil Roy
Ann. Probab. 38(2): 770-793 (March 2010). DOI: 10.1214/09-AOP495


We consider a point process sequence induced by a stationary symmetric α-stable (0<α<2) discrete parameter random field. It is easy to prove, following the arguments in the one-dimensional case in [Stochastic Process. Appl. 114 (2004) 191–210], that if the random field is generated by a dissipative group action then the point process sequence converges weakly to a cluster Poisson process. For the conservative case, no general result is known even in the one-dimensional case. We look at a specific class of stable random fields generated by conservative actions whose effective dimensions can be computed using the structure theorem of finitely generated Abelian groups. The corresponding point processes sequence is not tight, and hence needs to be properly normalized in order to ensure weak convergence. This weak limit is computed using extreme value theory and some counting techniques.


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Parthanil Roy. "Ergodic theory, Abelian groups and point processes induced by stable random fields." Ann. Probab. 38 (2) 770 - 793, March 2010.


Published: March 2010
First available in Project Euclid: 9 March 2010

zbMATH: 1204.60037
MathSciNet: MR2642891
Digital Object Identifier: 10.1214/09-AOP495

Primary: 60G55
Secondary: 37A40 , 60G60 , 60G70

Keywords: ergodic theory , Extreme value theory , group action , point process , Random field , random measure , Stable process , weak convergence

Rights: Copyright © 2010 Institute of Mathematical Statistics

Vol.38 • No. 2 • March 2010
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