Ergodic Theory, Abelian Groups, and Point Processes Induced by Stable Random Fields

Details Ergodic Theory, Abelian Groups, and Point Processes Induced by Stable Random Fields Ergodic Theory, Abelian Groups, and Point Processes Induced by Stable Random Fields

2007-12-05

arxiv.org/abs/0712.0688v2

Mathematics

Probability

To appear in the Annals of Probability. Some typos are fixed and Remark 4.3 (due to Jan Rosinski) is added in the second versi

Scientific paper

We consider a point process sequence induced by a stationary symmetric alpha-stable (0 < alpha < 2) discrete parameter random field. It is easy to prove, following the arguments in the one-dimensional case in Resnick and Samorodnitsky (2004), 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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