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We study the point process $W$ in $\mathbb{R}^d$ obtained by adding anindependent Gaussian vector to each point in $\mathbb{Z}^d$. Our main concernis the asymptotic size of fluctuations of the linear statistics in the largevolume limit, defined as  \[  N(h,R) = \sum_{w\in W} h\left(\frac{w}{R}\right),  \]  where $h\in \left(L^1\cap L^2\right)(\mathbb{R}^d)$ is a test function and$R\to \infty$. We will also consider the stationary counter-part of the process$W$, obtained by adding to all perturbations a random vector which is uniformlydistributed on $[0,1]^d$ and is independent of all the Gaussians. We focus ontwo main examples of interest, when the test function $h$ is either smooth oris an indicator function of a convex set with a smooth boundary whose curvaturedoes not vanish.

Fluctuations of linear statistics for Gaussian perturbations of the lattice $\mathbb{Z}^d$