  We establish a multivariate empirical process central limit theorem for
stationary $\R^d$-valued stochastic processes $(X_i)_{i\geq 1}$ under very weak
conditions concerning the dependence structure of the process. As an
application we can prove the empirical process CLT for ergodic torus
automorphisms. Our results also apply to Markov chains and dynamical systems
having a spectral gap on some Banach space of functions. Our proof uses a
multivariate extension of the techniques introduced by Dehling, Durieu and
Voln\'y \cite{DehDurVol09} in the univariate case. As an important technical
ingredient, we prove a $(2p)$th moment bound for partial sums in multiple
mixing systems.
