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arXiv · 1804.09929

On the CLT for rotations and BV functions

Jean-Pierre Conze·Stéphane Le Borgne
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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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Jean-Pierre Conze, Stéphane Le Borgne. 2018-04-26. On the CLT for rotations and BV functions. https://arxiv.org/abs/1804.09929

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