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 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 ℝ^d, and ξ=(ξ_t)_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 β≤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 $\mathscr{S}'(\mathbb {R}^{d})$-valued processes.