The Annals of Probability
- Ann. Probab.
- Volume 38, Number 2 (2010), 770-793.
Ergodic theory, Abelian groups and point processes induced by stable random fields
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.
Ann. Probab., Volume 38, Number 2 (2010), 770-793.
First available in Project Euclid: 9 March 2010
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
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. https://projecteuclid.org/euclid.aop/1268143532