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We prove limit theorems for rescaled occupation time fluctuations of a(d,alpha,beta)-branching particle system (particles moving in R^d according toa spherically symmetric alpha-stable Levy process, (1+beta)-branching,0<beta<1, uniform Poisson initial state), in the cases of critical dimension,d=alpha(1+beta)/beta, and large dimensions, d>alpha(1+beta)/beta. Thefluctuation processes are continuous but their limits are stable processes withindependent increments, which have jumps. The convergence is in the sense offinite-dimensional distributions, and also of space-time random fields(tightness does not hold in the usual Skorohod topology). The results are insharp contrast with those for intermediate dimensions, alpha/beta < d <d(1+beta)/beta, where the limit process is continuous and has long rangedependence (this case is studied by Bojdecki et al, 2005). The limit process ismeasure-valued for the critical dimension, and S'(R^d)-valued for largedimensions. We also raise some questions of interpretation of the differenttypes of dimension-dependent results obtained in the present and previouspapers in terms of properties of the particle system.Comment: 18 page

Occupation time fluctuations of an infinite-variance branching system in large dimensions