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 $\mathbb {R}^d$ 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 $\mathbb {R}^d$, and $ξ=(ξ_t)_{t\geq0}$ 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 $β\leq d/(d+α)$, 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 $(1+β-d/(d+α))d/α$ for the second one. The convergence proofs use techniques of $\mathcal{S}&#39;(\mathbb {R}^d)$-valued processes.