  We study the point process $W$ in $\mathbb{R}^d$ obtained by adding an
independent Gaussian vector to each point in $\mathbb{Z}^d$. Our main concern
is the asymptotic size of fluctuations of the linear statistics in the large
volume 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 uniformly
distributed on $[0,1]^d$ and is independent of all the Gaussians. We focus on
two main examples of interest, when the test function $h$ is either smooth or
is an indicator function of a convex set with a smooth boundary whose curvature
does not vanish.
