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 $(d,\alpha ,\beta )$-branching particle system. It consists of particles moving according to a symmetric $\alpha$-stable motion in ${\mathbb{R}}^d$. The branching law is in the domain of attraction of a (1+$\beta$)-stable law and the initial condition is an equilibrium random measure for the system (defined below). In the paper we treat separately the cases of intermediate $\alpha /\beta <d<(1+\beta )\alpha /\beta$, critical $d=(1+\beta )\alpha /\beta$ and large $d>(1+\beta )\alpha /\beta$ 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.