  In this paper we extend the refined second-order Poincar\'e inequality for
Poisson functionals from a one-dimensional to a multi-dimensional setting. Its
proof is based on a multivariate version of the Malliavin-Stein method for
normal approximation on Poisson spaces. We also present an application to
partial sums of vector-valued functionals of heavy-tailed moving averages. The
extension allows a functional with multivariate arguments, i.e. multiple moving
averages and also multivariate values of the functional. Such a set-up has
previously not been explored in the framework of stable moving average
processes. It can potentially capture probabilistic properties which cannot be
described solely by the one-dimensional marginals, but instead require the
joint distribution.
