A Long Range Dependence Stable Process and an Infinite Variance Branching System

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-TRS425, RePAd Working Paper Series from Département des sciences administratives, UQO

Abstract: We prove a functional limit theorem for the rescaled occupation time fluctuations of a (d, , )- branching particle system (particles moving in Rd according to a symmetric -stable L´evy process, branching law in the domain of attraction of a (1 + )-stable law, 0 < < 1, uniform Poisson initial state) in the case of intermediate dimensions, / < d < (1 + )/ . The limit is a process of the form K, where K is a constant,  is the Lebesgue measure on Rd, and  = (t)t0 is a (1+ )-stable process which has long range dependence. There are two long range dependence regimes, one for all > d/(d + ), which coincides with the case of finite variance branching ( = 1), and another one for  d/(d + ), where the long range dependence depends on the value of . The long range dependence is characterized by a dependence exponent  which describes the asymptotic behavior of the codierence of increments of  on intervals far apart, and which is d/ for the first case and (1 + - d/(d + ))d/ for the second one. The convergence proofs use techniques of S0(Rd)-valued processes.

Keywords: Branching particle system; occupation time fluctuation; functional limit theorem; stable process; long range dependence. (search for similar items in EconPapers)
JEL-codes: C10 C40 (search for similar items in EconPapers)
Date: Written 2005-11-14
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