The Annals of Probability

Ergodic theory, Abelian groups and point processes induced by stable random fields

Parthanil Roy

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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.

Article information

Ann. Probab., Volume 38, Number 2 (2010), 770-793.

First available in Project Euclid: 9 March 2010

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Zentralblatt MATH identifier

Primary: 60G55: Point processes
Secondary: 60G60: Random fields 60G70: Extreme value theory; extremal processes 37A40: Nonsingular (and infinite-measure preserving) transformations

Stable process random field point process random measure weak convergence extreme value theory ergodic theory group action


Roy, Parthanil. Ergodic theory, Abelian groups and point processes induced by stable random fields. Ann. Probab. 38 (2010), no. 2, 770--793. doi:10.1214/09-AOP495.

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