  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.
