  We consider two independent Gaussian processes that admit a representation in
terms of a stochastic integral of a deterministic kernel with respect to a
standard Wiener process. In this paper we construct two families of processes,
from a unique Poisson process, the finite dimensional distributions of which
converge in law towards the finite dimensional distributions of the two
independent Gaussian processes. As an application of this result we obtain
families of processes that converge in law towards fractional Brownian motion
and sub-fractional Brownian motion.
