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 R d with symmetric α -stable motion and α < d < 2 α , 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 < α , which involves fractional Brownian motion. We use a space–time random field approach.