A Long Range Dependence Stable Process and an Infinite Variance Branching System

We prove a functional limit theorem for the rescaled occupation time fluctuations of a (d, α, β)branching particle system (particles moving in R 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, and ξ = (ξt)t≥0 is a (1+β)-stable process which has long range dependence. There are two long range dependence regimes, one for all 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 (1 + β − d/(d + α))d/α for the second one. The convergence proofs use techniques of S ′(R)-valued processes. AMS subject classifications: 60F17, 60J80, 60G18, 60G52.