We construct fractional Brownian motion, sub-fractional Brownian motion and negative sub-fractional Brownian motion by means of limiting procedures applied to some particle systems. These processes are obtained for full ranges of Hurst parameter.

We prove a functional limit theorem for the rescaled occupation time fluctuations of a (d, , )- branching particle system (particles moving in Rd according to a symmetric -stable L´evy process, branching law in the domain of attraction of a (1 + )-stable law, 0 < < 1, uniform Poisson initial state) in the case of intermediate dimensions, / < d < (1 + )/. The limit is a process of the form K, where K is a constant, is the Lebesgue measure on Rd, and = (t)t0 is a (1+)-stable process which has long range dependence. There are two long range dependence regimes, one for all > d/(d + ), which coincides with...</<>

We prove limit theorems for rescaled occupation time fluctuations of a (d, , )-branching particle system (particles moving in Rd according to a spherically symmetric -stable L´evy process, (1 + )- branching, 0 < < 1, uniform Poisson initial state), in the cases of critical dimension, d = (1+)/, and large dimensions, d > (1 + )/. The fluctuation processes are continuous but their limits are stable...</<>