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 with symmetric [alpha]-stable motion in the cases of critical and large dimensions, d=2[alpha] and d>2[alpha]. 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, [alpha]<d<2[alpha], 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=[alpha] and d>[alpha].