Occupation Time Fluctuations of an Infinite Variance Branching System in Large Dimensions

T. Bojdecki (), Luis G. Gorostiza () and A. Talarczyk ()
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T. Bojdecki: Institute of Mathematics, University of Warsaw
Luis G. Gorostiza: Departamento de Mathematicas, Centro de Investigacion y de Estudios Avanzados, LRSP
A. Talarczyk: Institute of Mathematics, University of Warsaw

No lrsp-TRS426, RePAd Working Paper Series from Département des sciences administratives, UQO

Abstract: We prove limit theorems for rescaled occupation time fluctuations of a (d, , )-branching particle system (particles moving in Rd according to a spherically symmetric -stable L´evy process, (1 + )- branching, 0 < < 1, uniform Poisson initial state), in the cases of critical dimension, d = (1+ )/ , and large dimensions, d > (1 + )/ . The fluctuation processes are continuous but their limits are stable processes with independent increments, which have jumps. The convergence is in the sense of finite-dimensional distributions, and also of space-time random fields (tightness does not hold in the usual Skorohod topology). The results are in sharp contrast with those for intermediate dimensions, / < d < d(1+ )/ , where the limit process is continuous and has long range dependence (this case is studied by Bojdecki et al, 2005c). The limit process is measure-valued for the critical dimension, and S0(Rd)-valued for large dimensions. We also raise some questions of interpretation of the dierent types of dimension-dependent results obtained in the present and previous papers in terms of properties of the particle system.

Keywords: Branching particle system; critical and large dimensions; limit theorem; occupation time fluctuation; stable process. (search for similar items in EconPapers)
JEL-codes: C10 C40 (search for similar items in EconPapers)
New Economics Papers: this item is included in nep-ecm
Date: 2005-11-14

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