OCCUPATION TIME FLUCTUATION LIMITS OF INFINITE VARIANCE EQUILIBRIUM BRANCHING SYSTEMS

Abstract

We establish limit theorems for the fluctuations of the rescaled occupation time of a (d, α, β)-branching particle system. It consists of particles moving according to a symmetric α-stable motion in ℝd. The branching law is in the domain of attraction of a (1 + β)-stable law and the initial condition is the equilibrium random measure for the system (defined below). In the paper we treat separately the cases of intermediate α/β < d < (1 + β)α/β, critical d = (1 + β)α/β and large d > (1 + β)α/β dimensions. In the most interesting case of intermediate dimensions we obtain a version of a fractional stable motion. The long-range dependence structure of this process is also studied. Contrary to this case, limit processes in critical and large dimensions have independent increments.