On the CLT for rotations and BV functions
Annales mathématiques Blaise Pascal, Tome 29 (2022) no. 1, pp. 51-97.

Let xx+αmod1 be a rotation on the circle and let φ be a step function. We denote by φn(x) the corresponding ergodic sums j=0n-1φ(x+jα). 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 φn/φn2 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 qk, where (qk) is the sequence of denominators of α. Another important point is the control of the variance φn22 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 :
DOI : 10.5802/ambp.407
Classification : 11A55, 37E10, 60F05
Mots-clés : irrational rotations, central limit theorem

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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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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