  Let $\rho$ be a probability measure on $\mathrm{SL}\_d(\mathbb{Z})$ and
consider the random walk defined by $\rho$ on the torus $\mathbb{T}^d =
\mathbb{R}^d/\mathbb{Z}^d$. Bourgain, Furmann, Lindenstrauss and Mozes proved
that under an assumption on the group generated by the support of $$\rho$$, the
random walk starting at any irrational point equidistributes in the torus. In
this article, we study the central limit theorem and the law of the iterated
logarithm for this walk starting at some point having good diophantine
properties.
