  We prove a functional limit theorem for the rescaled occupation time
fluctuations of a $(d,\alpha,\beta)$-branching particle system [particles
moving in $\mathbb {R}^d$ according to a symmetric $\alpha$-stable L\'{e}vy
process, branching law in the domain of attraction of a $(1+\beta)$-stable law,
$0<\beta<1$, uniform Poisson initial state] in the case of intermediate
dimensions, $\alpha/\beta<d<\alpha(1+\beta)/\beta$. The limit is a process of
the form $K\lambda\xi$, where $K$ is a constant, $\lambda$ is the Lebesgue
measure on $\mathbb {R}^d$, and $\xi=(\xi_t)_{t\geq0}$ is a $(1+\beta)$-stable
process which has long range dependence. For $\alpha<2$, there are two long
range dependence regimes, one for $\beta>d/(d+\alpha)$, which coincides with
the case of finite variance branching $(\beta=1)$, and another one for
$\beta\leq d/(d+\alpha)$, where the long range dependence depends on the value
of $\beta$. The long range dependence is characterized by a dependence exponent
$\kappa$ which describes the asymptotic behavior of the codifference of
increments of $\xi$ on intervals far apart, and which is $d/\alpha$ for the
first case (and for $\alpha=2$) and $(1+\beta-d/(d+\alpha))d/\alpha$ for the
second one. The convergence proofs use techniques of $\mathcal{S}'(\mathbb
{R}^d)$-valued processes.
