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 $(d,\alpha ,\beta )$-branching particle system [particles moving in ${\mathbb{R}}^d$ according to a symmetric $\alpha$-stable Lévy 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, $\alpha /\beta <d<\alpha (1+\beta )/\beta$. The limit is a process of the form $K\lambda \xi$, where $K$ is a constant, $\lambda$ is the Lebesgue measure on ${\mathbb{R}}^d$, and $\xi =({\xi }_t{)}_{t\ge 0}$ is a $(1+\beta )$-stable process which has long range dependence. For $\alpha <2$, there are two long range dependence regimes, one for $\beta >d/(d+\alpha )$, which coincides with the case of finite variance branching $(\beta =1)$, and another one for $\beta \le d/(d+\alpha )$, 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 ${\mathcal{S}}^{\prime }({\mathbb{R}}^d)$-valued processes.

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

References in corpus (2)

• Occupation time fluctuations of an infinite variance branching system in large dimensions
• Occupation time fluctuations of Poisson and equilibrium finite variance branching systems

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