On the CLT for rotations and BV functions
Annales mathématiques Blaise Pascal, Volume 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 φ n (x) the corresponding ergodic sums j=0 n-1 φ(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 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.

Published online:
DOI: 10.5802/ambp.407
Classification: 11A55, 37E10, 60F05
Keywords: irrational rotations, central limit theorem
Jean-Pierre Conze 1; Stéphane Le Borgne 1

1 Univ Rennes, CNRS, IRMAR - UMR 6625, F-35000 Rennes, France
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
@article{AMBP_2022__29_1_51_0,
     author = {Jean-Pierre Conze and St\'ephane Le Borgne},
     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 = {https://ambp.centre-mersenne.org/articles/10.5802/ambp.407/}
}
TY  - JOUR
AU  - Jean-Pierre Conze
AU  - Stéphane Le Borgne
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  - https://ambp.centre-mersenne.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 Jean-Pierre Conze
%A Stéphane Le Borgne
%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 https://ambp.centre-mersenne.org/articles/10.5802/ambp.407/
%R 10.5802/ambp.407
%G en
%F AMBP_2022__29_1_51_0
Jean-Pierre Conze; Stéphane Le Borgne. 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. https://ambp.centre-mersenne.org/articles/10.5802/ambp.407/

[1] József Beck Randomness of the square root of 2 and the giant leap I, Period. Math. Hung., Volume 60 (2010) no. 2, pp. 137-242 | DOI | MR | Zbl

[2] József Beck Probabilistic Diophantine approximation, Randomness in lattice point counting, Springer Monographs in Mathematics, Springer, 2014

[3] Michael Bromberg; Corinna Ulcigrai A temporal central limit theorem for real-valued cocycles over rotations, Ann. Inst. Henri Poincaré, Probab. Stat., Volume 54 (2018) no. 4, pp. 2304-2334 | MR | Zbl

[4] Robert Burton; Manfred Denker On the central limit theorem for dynamical systems, Trans. Am. Math. Soc., Volume 302 (1987) no. 2, pp. 715-726 | DOI | MR | Zbl

[5] Jean-Pierre Conze Recurrence, ergodicity and invariant measures for cocycles over a rotation, Ergodic theory (Contemporary Mathematics), Volume 485, American Mathematical Society, 2009, pp. 45-70 | DOI | MR | Zbl

[6] Jean-Pierre Conze; Eugene Gutkin On recurrence and ergodicity for geodesic flows on non-compact periodic polygonal surfaces, Ergodic Theory Dyn. Syst., Volume 32 (2012) no. 2, pp. 491-515 | DOI | MR | Zbl

[7] Jean-Pierre Conze; Stefano Isola; Stéphane Le Borgne Diffusive behaviour of ergodic sums over rotations, Stoch. Dyn., Volume 19 (2019) no. 2, 1950016, 26 pages | Zbl

[8] Jean-Pierre Conze; Stéphane Le Borgne On the CLT for rotations and BV functions, C. R. Math. Acad. Sci. Paris, Volume 357 (2019) no. 2, pp. 212-215 | DOI | MR | Zbl

[9] Thierry De la Rue; Stéphane Ladouceur; Goran Peskir; Michel Weber On the central limit theorem for aperiodic dynamical systems and applications, Teor. Jmovirn. Mat. Stat., Volume 57 (1997), pp. 140-159 | Zbl

[10] Dmitry Dolgopyat; Omri Sarig Temporal distributional limit theorems for dynamical systems, J. Stat. Phys., Volume 166 (2017) no. 3-4, pp. 680-713 | DOI | MR | Zbl

[11] Joseph L. Doob Stochastic Processes, John Wiley & Sons, 1953

[12] William Feller An introduction to probability theory and its application, John Wiley & Sons, 1966

[13] Mélanie Guenais; François Parreau Valeurs propres de transformations liées aux rotations irrationnelles et aux fonctions en escalier (2006) (https://arxiv.org/abs/math/0605250)

[14] Godfrey H. Hardy; John E. Littlewood Some problems of diophantine approximation: a series of cosecants, Bull. Calcutta Math. Soc., Volume 20 (1930), pp. 251-266 | Zbl

[15] François Huveneers Subdiffusive behavior generated by irrational rotations, Ergodic Theory Dyn. Syst., Volume 29 (2009), pp. 1217-1233 | DOI | MR | Zbl

[16] Aleksandr Khinchin Continued Fractions, P. Noordhoff, 1963

[17] Bruce P. Kitchens Symbolic dynamics. One-sided, two-sided and countable state Markov shifts, Universitext, Springer, 1998

[18] Michael T. Lacey On central limit theorems, modulus of continuity and Diophantine type for irrational rotations, J. Anal. Math., Volume 61 (1993), pp. 47-59 | DOI | MR | Zbl

[19] Serge Lang Introduction to Diophantine Approximations, Addison-Wesley Publishing Group, 1966 | Numdam

[20] Pascal Lezaud Chernoff-type bound for finite Markov chains, Ann. Appl. Probab., Volume 8 (1998) no. 3, pp. 849-867 | MR | Zbl

[21] Pierre Liardet; Dalibor Volný Sums of continuous and differentiable functions in dynamical systems, Isr. J. Math., Volume 98 (1997), pp. 29-60 | DOI | MR | Zbl

[22] Alexander Ostrowski Bemerkungen zur Theorie der Diophantischen Approximationen, Abh. Math. Semin. Univ. Hamb., Volume 1 (1922), pp. 77-99 | DOI | MR | Zbl

[23] Jean-Paul Thouvenot; Benjamin Weiss Limit laws for ergodic processes, Stoch. Dyn., Volume 12 (2012) no. 1, 1150012, 9 pages | MR | Zbl

Cited by Sources: