On the CLT for rotations and BV functions

Conze, Jean-Pierre; Le Borgne, Stéphane

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

Let $x\mapsto x+\alpha mod1$ be a rotation on the circle and let $\phi$ be a step function. We denote by ${\phi }_n\left(x\right)$ the corresponding ergodic sums ${\sum }_{j=0}^{n-1}\phi (x+j\alpha )$. For a class of irrational rotations (containing the class with bounded partial quotients) and under a Diophantine condition on the discontinuity points of $\phi$, we show that ${\phi }_n/{\parallel {\phi }_n\parallel }_2$ 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 $\left(q_k\right)$ is the sequence of denominators of $\alpha$. Another important point is the control of the variance $\parallel {\phi }_n{\parallel }_2^2$ for $n$ belonging to a large set of integers. When $\alpha$ is a quadratic irrational, the size of this set can be precisely estimated.

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

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

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

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

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

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

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

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

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

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

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

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

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

[18] Lacey, Michael T. 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] Lang, Serge Introduction to Diophantine Approximations, Addison-Wesley Publishing Group, 1966 | Numdam

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

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

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

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