We prove limit theorems for rescaled occupation time fluctuations of a (d, α, β)-branching particle system (particles moving in ℝd according to a spherically symmetric α-stable Lévy process, (1+β)-branching, 0<β<1, uniform Poisson initial state), in the cases of critical dimension, d=α(1+β)/β, and large dimensions, d>α(1+β)/β. The fluctuation processes are continuous but their limits are stable processes with independent increments, which have jumps. The convergence is in the sense of finite-dimensional distributions, and also of space-time random fields (tightness does not hold in the usual Skorohod topology). The results are in sharp contrast with those for intermediate dimensions, α/β