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

We give a functional limit theorem for the fluctuations of the resealed occupation time process of a critical branching particle system in R-d 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. (c) 2005 Elsevier B.V. All rights reserved.