Let be a rotation on the circle and let be a step function. We denote by the corresponding ergodic sums . For a class of irrational rotations (containing the class with bounded partial quotients) and under a Diophantine condition on the discontinuity points of , we show that is asymptotically Gaussian for in a set of density 1. The proof is based on decorrelation inequalities for the ergodic sums taken at times , where is the sequence of denominators of . Another important point is the control of the variance for belonging to a large set of integers. When is a quadratic irrational, the size of this set can be precisely estimated.
Keywords: irrational rotations, central limit theorem
@article{AMBP_2022__29_1_51_0, author = {Conze, Jean-Pierre and Le Borgne, St\'ephane}, 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}, volume = {29}, number = {1}, year = {2022}, doi = {10.5802/ambp.407}, language = {en}, url = {http://www.numdam.org/articles/10.5802/ambp.407/} }
TY - JOUR AU - Conze, Jean-Pierre AU - Le Borgne, Stéphane TI - On the CLT for rotations and BV functions JO - Annales mathématiques Blaise Pascal PY - 2022 SP - 51 EP - 97 VL - 29 IS - 1 PB - Université Clermont Auvergne, Laboratoire de mathématiques Blaise Pascal UR - http://www.numdam.org/articles/10.5802/ambp.407/ DO - 10.5802/ambp.407 LA - en ID - AMBP_2022__29_1_51_0 ER -
%0 Journal Article %A Conze, Jean-Pierre %A Le Borgne, Stéphane %T On the CLT for rotations and BV functions %J Annales mathématiques Blaise Pascal %D 2022 %P 51-97 %V 29 %N 1 %I Université Clermont Auvergne, Laboratoire de mathématiques Blaise Pascal %U http://www.numdam.org/articles/10.5802/ambp.407/ %R 10.5802/ambp.407 %G en %F AMBP_2022__29_1_51_0
Conze, Jean-Pierre; Le Borgne, Stéphane. On the CLT for rotations and BV functions. Annales mathématiques Blaise Pascal, Volume 29 (2022) no. 1, pp. 51-97. doi : 10.5802/ambp.407. http://www.numdam.org/articles/10.5802/ambp.407/
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