Limit theorems for occupation time fluctuations of branching systems I: Long-range dependence

We give a functional limit theorem for the fluctuations of the rescaled occupation time process of a critical branching particle system in with symmetric [alpha]-stable motion and [alpha]<d<2[alpha], which leads to a long-range dependence process involving sub-fractional Brownian motion. We also give an analogous result for the system without branching and d<[alpha], which involves fractional Brownian motion. We use a space-time random field approach.