Filtered Brownian motions as weak limit of filtered Poisson processes

The main result of this paper is a limit theorem which shows the convergence in law, on a Hölderian space, of filtered Poisson processes (a class of processes which contains shot noise process) to filtered Brownian motion (a class of processes which contains fractional Brownian motion) when the intensity of the underlying Poisson process is increasing. We apply the theory of convergence of Hilbert space valued semimartingales and use a radonification result.