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We prove a functional limit theorem for the rescaled occupation timefluctuations of a $(d,\alpha,\beta)$-branching particle system [particlesmoving in $\mathbb {R}^d$ according to a symmetric $\alpha$-stable L\'{e}vyprocess, branching law in the domain of attraction of a $(1+\beta)$-stable law,$0<\beta<1$, uniform Poisson initial state] in the case of intermediatedimensions, $\alpha/\beta<d<\alpha(1+\beta)/\beta$. The limit is a process ofthe form $K\lambda\xi$, where $K$ is a constant, $\lambda$ is the Lebesguemeasure on $\mathbb {R}^d$, and $\xi=(\xi_t)_{t\geq0}$ is a $(1+\beta)$-stableprocess which has long range dependence. For $\alpha<2$, there are two longrange dependence regimes, one for $\beta>d/(d+\alpha)$, which coincides withthe case of finite variance branching $(\beta=1)$, and another one for$\beta\leq d/(d+\alpha)$, where the long range dependence depends on the valueof $\beta$. The long range dependence is characterized by a dependence exponent$\kappa$ which describes the asymptotic behavior of the codifference ofincrements of $\xi$ on intervals far apart, and which is $d/\alpha$ for thefirst case (and for $\alpha=2$) and $(1+\beta-d/(d+\alpha))d/\alpha$ for thesecond one. The convergence proofs use techniques of $\mathcal{S}'(\mathbb{R}^d)$-valued processes.

A long range dependence stable process and an infinite variance branching system