Annals of Probability

A long range dependence stable process and an infinite variance branching system

Tomasz Bojdecki, Luis G. Gorostiza, and Anna Talarczyk

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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.

Article information

Ann. Probab., Volume 35, Number 2 (2007), 500-527.

First available in Project Euclid: 30 March 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60F17: Functional limit theorems; invariance principles
Secondary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60G18: Self-similar processes 60G52: Stable processes

Branching particle system occupation time fluctuation functional limit theorem stable process long range dependence


Bojdecki, Tomasz; Gorostiza, Luis G.; Talarczyk, Anna. A long range dependence stable process and an infinite variance branching system. Ann. Probab. 35 (2007), no. 2, 500--527. doi:10.1214/009117906000000737.

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