Limit theorems for occupation time fluctuations of branching systems II: Critical and large dimensions

We give functional limit theorems for the fluctuations of the rescaled occupation time process of a critical branching particle system in R d with symmetric α -stable motion in the cases of critical and large dimensions, d = 2 α and d > 2 α . In a previous paper [T. Bojdecki, L.G. Gorostiza, A. Talarczyk, Limit theorems for occupation time fluctuations of branching systems I: long-range dependence, Stochastic Process. Appl., this issue.] we treated the case of intermediate dimensions, α < d < 2 α , which leads to a long-range dependence limit process. In contrast, in the present cases the limits are generalized Wiener processes. We use the same space–time random field method of the previous paper, the main difference being that now the tightness requires a new approach and the proofs are more difficult. We also give analogous results for the system without branching in the cases d = α and d > α .