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On the CLT for rotations and BV functions

Abstract : 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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Submitted on : Monday, January 10, 2022 - 9:37:08 AM
Last modification on : Friday, May 20, 2022 - 9:04:52 AM

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Jean-Pierre Conze, Stéphane Le Borgne. On the CLT for rotations and BV functions. Comptes Rendus. Mathématique, Académie des sciences (Paris), 2019, 357 (2), pp.212-215. ⟨10.1016/j.crma.2019.01.008⟩. ⟨hal-01777584v4⟩

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