  This paper is devoted to the study of the stochastic properties of dynamical
systems preserving an infinite measure. More precisely we prove central limit
theorems for Birkhoff sums of observables of $\mathbb{Z}^2$-extensions of
dynamical systems (satisfying some nice spectral properties). The motivation of
our paper is the $\mathbb{Z}^2$-periodic Lorentz process for which we establish
a functional central limit theorem for H\"older continuous observables (in
discrete time as well as in continuous time).
