On the CLT for rotations and BV functions
Comptes Rendus. Math{\'e}matique, Acad{\'e}mie des sciences (Paris), 2019, 357 (2), pp.212-215. \&\#x27E8;10.1016/j.crma.2019.01.008\&\#x27E9 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...
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Zusammenfassung: | Comptes Rendus. Math{\'e}matique, Acad{\'e}mie des sciences
(Paris), 2019, 357 (2), pp.212-215.
\&\#x27E8;10.1016/j.crma.2019.01.008\&\#x27E9 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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DOI: | 10.48550/arxiv.1804.09929 |