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

TL;DR

This paper investigates the statistical properties, specifically the central limit theorem and law of the iterated logarithm, for a linear random walk on the torus driven by a probability measure on SL_d(Z), focusing on points with good diophantine properties.

Contribution

It extends previous work by establishing CLT and LIL results for the walk starting at points with certain diophantine conditions, under group assumptions.

Findings

Proves CLT for the linear random walk on the torus.

Establishes the law of the iterated logarithm for the walk.

Demonstrates results for points with good diophantine properties.

Abstract

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.