## The Annals of Probability

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

#### 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 -valued processes.

#### Article information

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

Dates
First available in Project Euclid: 30 March 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1175287752

Digital Object Identifier
doi:10.1214/009117906000000737

Mathematical Reviews number (MathSciNet)
MR2308586

Zentralblatt MATH identifier
1121.60028

#### Citation

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. https://projecteuclid.org/euclid.aop/1175287752

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