Central limit theorem and law of the iterated logarithm for the linear random walk on the torus

Let $\rho$ be a probability measure on $\mathrm{S}\mathrm{L}\mathrm{\_}d(\mathbb{Z})$ 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 \text{,}$$ 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.