On the CLT for rotations and BV functions

Jean-Pierre Conze; Stéphane Le Borgne

doi:10.5802/ambp.407

Jean-Pierre Conze ¹ ; Stéphane Le Borgne ¹

¹ Univ Rennes, CNRS, IRMAR - UMR 6625, F-35000 Rennes, France

Annales mathématiques Blaise Pascal, Tome 29 (2022) no. 1, pp. 51-97.

Résumé

Let $x \mapsto x + α mod 1$ be a rotation on the circle and let $φ$ be a step function. We denote by $φ_{n} (x)$ the corresponding ergodic sums $\sum_{j = 0}^{n - 1} φ (x + j α)$ . For a class of irrational rotations (containing the class with bounded partial quotients) and under a Diophantine condition on the discontinuity points of $φ$ , we show that $φ_{n} / {∥ φ_{n} ∥}_{2}$ is asymptotically Gaussian for $n$ in a set of density 1. The proof is based on decorrelation inequalities for the ergodic sums taken at times $q_{k}$ , where $(q_{k})$ is the sequence of denominators of $α$ . Another important point is the control of the variance $∥ φ_{n} ∥_{2}^{2}$ for $n$ belonging to a large set of integers. When $α$ is a quadratic irrational, the size of this set can be precisely estimated.

Publié le : 2022-09-23

DOI : 10.5802/ambp.407

Classification : 11A55, 37E10, 60F05
Mots clés : irrational rotations, central limit theorem

Affiliations des auteurs :

Jean-Pierre Conze ¹ ; Stéphane Le Borgne ¹

¹ Univ Rennes, CNRS, IRMAR - UMR 6625, F-35000 Rennes, France

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     title = {On the {CLT} for rotations and {BV} functions},
     journal = {Annales math\'ematiques Blaise Pascal},
     pages = {51--97},
     publisher = {Universit\'e Clermont Auvergne, Laboratoire de math\'ematiques Blaise Pascal},
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Jean-Pierre Conze; Stéphane Le Borgne. On the CLT for rotations and BV functions. Annales mathématiques Blaise Pascal, Tome 29 (2022) no. 1, pp. 51-97. doi : 10.5802/ambp.407. https://ambp.centre-mersenne.org/articles/10.5802/ambp.407/

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