On the CLT for rotations and BV functions

Jean-Pierre Conze (1) , Stéphane Le Borgne (1)

1 IRMAR - Institut de Recherche Mathématique de Rennes

Let $x \mapsto x+ \alpha$ be a rotation on the circle and let $\varphi$ be a step function. We denote by $\varphi_n (x)$ the corresponding ergodic sums $\sum_{j=0}^{n-1} \varphi(x+j \alpha)$. Under an assumption on $\alpha$, for example when $\alpha$ has bounded partial quotients, and a Diophantine condition on the discontinuity points of $\varphi$, we show that $\varphi_n/\|\varphi_n\|_2$ is asymptotically Gaussian for $n$ in a set of density 1. The method is based on decorrelation inequalities for the ergodic sums taken at times $q_k$, where the $q_k$'s are the denominators of $\alpha$.

Domaines: Systèmes dynamiques [math.DS]; Probabilités [math.PR]

Jean-Pierre Conze, Stéphane Le Borgne. On the CLT for rotations and BV functions. Comptes Rendus. Mathématique, 2019, 357 (2), pp.212-215. ⟨10.1016/j.crma.2019.01.008⟩. ⟨hal-01777584v3⟩