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March 2007 A long range dependence stable process and an infinite variance branching system
Tomasz Bojdecki, Luis G. Gorostiza, Anna Talarczyk
Ann. Probab. 35(2): 500-527 (March 2007). DOI: 10.1214/009117906000000737

Abstract

We prove a functional limit theorem for the rescaled occupation time fluctuations of a (d, α, β)-branching particle system [particles moving in ℝd according to a symmetric α-stable Lévy process, branching law in the domain of attraction of a (1+β)-stable law, 0<β<1, uniform Poisson initial state] in the case of intermediate dimensions, α/β<d<α(1+β)/β. The limit is a process of the form Kλξ, where K is a constant, λ is the Lebesgue measure on ℝd, and ξ=(ξt)t≥0 is a (1+β)-stable process which has long range dependence. For α<2, there are two long range dependence regimes, one for β>d/(d+α), which coincides with the case of finite variance branching (β=1), and another one for βd/(d+α), where the long range dependence depends on the value of β. The long range dependence is characterized by a dependence exponent κ which describes the asymptotic behavior of the codifference of increments of ξ on intervals far apart, and which is d/α for the first case (and for α=2) and (1+βd/(d+α))d/α for the second one. The convergence proofs use techniques of $\mathscr{S}'(\mathbb {R}^{d})$-valued processes.

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Tomasz Bojdecki. Luis G. Gorostiza. Anna Talarczyk. "A long range dependence stable process and an infinite variance branching system." Ann. Probab. 35 (2) 500 - 527, March 2007. https://doi.org/10.1214/009117906000000737

Information

Published: March 2007
First available in Project Euclid: 30 March 2007

zbMATH: 1121.60028
MathSciNet: MR2308586
Digital Object Identifier: 10.1214/009117906000000737

Subjects:
Primary: 60F17
Secondary: 60G18 , 60G52 , 60J80

Keywords: Branching particle system , Functional limit theorem , Long range dependence , occupation time fluctuation , Stable process

Rights: Copyright © 2007 Institute of Mathematical Statistics

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Vol.35 • No. 2 • March 2007
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