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)
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 di erent types of dimension-dependent results obtained in the present and previous papers in terms of properties of the particle system.
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Paper provided by Département des sciences administratives, UQO in its series RePAd Working Paper Series with number
lrsp-TRS426.
Length: 18 pages Date of creation: 14 Nov 2005 Date of revision: Handle: RePEc:pqs:wpaper:112006
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