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

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

2005-11-30

arxiv.org/abs/math/0511739v2

Annals of Probability 2007, Vol. 35, No. 2, 500-527

Mathematics

Probability

Published at http://dx.doi.org/10.1214/009117906000000737 in the Annals of Probability (http://www.imstat.org/aop/) by the Ins

Scientific paper

10.1214/009117906000000737

We prove a functional limit theorem for the rescaled occupation time fluctuations of a $(d,\alpha,\beta)$-branching particle system [particles moving in $\mathbb {R}^d$ according to a symmetric $\alpha$-stable L\'{e}vy process, branching law in the domain of attraction of a $(1+\beta)$-stable law, $0<\beta<1$, uniform Poisson initial state] in the case of intermediate dimensions, $\alpha/\betad/(d+\alpha)$, which coincides with the case of finite variance branching $(\beta=1)$, and another one for $\beta\leq d/(d+\alpha)$, where the long range dependence depends on the value of $\beta$. The long range dependence is characterized by a dependence exponent $\kappa$ which describes the asymptotic behavior of the codifference of increments of $\xi$ on intervals far apart, and which is $d/\alpha$ for the first case (and for $\alpha=2$) and $(1+\beta-d/(d+\alpha))d/\alpha$ for the second one. The convergence proofs use techniques of $\mathcal{S}'(\mathbb {R}^d)$-valued processes.

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