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/

Bibliographie
Cité par

[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

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