Filtered Brownian motions as weak limit of filtered Poisson processes

Abstract : 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 semi-martingales and use
some result of radonification.