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Journal Articles Comptes Rendus. Mathématique Year : 2019

On the CLT for rotations and BV functions


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 $\alpha$.
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Dates and versions

hal-01777584 , version 1 (25-04-2018)
hal-01777584 , version 2 (13-06-2018)
hal-01777584 , version 3 (08-06-2021)
hal-01777584 , version 4 (10-01-2022)



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-01777584v4⟩
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