Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

A temporal central limit theorem for real-valued cocycles over rotations

Michael Bromberg and Corinna Ulcigrai

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Abstract

We consider deterministic random walks on the real line driven by irrational rotations, or equivalently, skew product extensions of a rotation by $\alpha$ where the skewing cocycle is a piecewise constant mean zero function with a jump by one at a point $\beta$. When $\alpha$ is badly approximable and $\beta$ is badly approximable with respect to $\alpha$, we prove a Temporal Central Limit theorem (in the terminology recently introduced by D. Dolgopyat and O. Sarig), namely we show that for any fixed initial point, the occupancy random variables, suitably rescaled, converge to a Gaussian random variable. This result generalizes and extends a theorem by J. Beck for the special case when $\alpha$ is quadratic irrational, $\beta$ is rational and the initial point is the origin, recently reproved and then generalized to cover any initial point using geometric renormalization arguments by Avila–Dolgopyat–Duryev–Sarig (Israel J. Math. 207 (2015) 653–717) and Dolgopyat–Sarig (J. Stat. Phys. 166 (2017) 680–713). We also use renormalization, but in order to treat irrational values of $\beta$, instead of geometric arguments, we use the renormalization associated to the continued fraction algorithm and dynamical Ostrowski expansions. This yields a suitable symbolic coding framework which allows us to reduce the main result to a CLT for non homogeneous Markov chains.

Résumé

On considère des marches aléatoires sur la droite réelle, engendrés par des rotations irrationnelles, ou, de manière équivalente, des produits croisés d’une rotation par un nombre réel $\alpha$, dont le cocycle est une fonction constante par morceaux de moyenne nulle admettant un saut de un à une singularité $\beta$. Si $\alpha$ est mal approché par des rationnels et $\beta$ n’est pas bien approché par l’orbite de $\alpha$, nous démontrons une version temporelle du Théorème de la Limite Centrale (ou un Temporal Central Limit theorem dans la terminologie qui a été introduite récemment par D. Dolgopyat et O. Sarig). Plus précisément, nous montrons que, pour chaque point initial fixé, les variables aléatoires d’occupation, proprement renormalisées, tendent vers une variable aléatoire de loi normale. Ce résultat généralise un théorème de J. Beck dans le cas sparticulier où $\alpha$ est un nombre irrationnel quadratique, $\beta$ est un nombre rationnel et le point initial est l’origine. Ce résultat de Beck a été montré avec de nouvelles méthodes et étendu par Avila–Dolgopyat–Duryev–Sarig (Israel J. Math. 207 (2015) 653–717) et Dolgopyat–Sarig (J. Stat. Phys. 166 (2017) 680–713) à l’aide d’une renormalisation géométrique. Dans ce papier, nous utilisons aussi la renormalisation, mais, au lieu d’avoir recours à un argument géométrique, nous proposons d’utiliser l’algorithme de fraction continue avec une version dynamique de l’expansion de Ostrowski. Cela nous donne un codage symbolique qui nous permet de réduire le résultat principal à un théorème de la limite centrale pour de chaînes de Markov non-homogènes.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 54, Number 4 (2018), 2304-2334.

Dates
Received: 26 May 2017
Revised: 2 October 2017
Accepted: 25 October 2017
First available in Project Euclid: 18 October 2018

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1539849800

Digital Object Identifier
doi:10.1214/17-AIHP872

Mathematical Reviews number (MathSciNet)
MR3865674

Zentralblatt MATH identifier
06996566

Subjects
Primary: 37A50: Relations with probability theory and stochastic processes [See also 60Fxx and 60G10] 37E10: Maps of the circle
Secondary: 11K06: General theory of distribution modulo 1 [See also 11J71]

Keywords
Limit theorems for dynamical systems Single orbit dynamics Skew-products over irrational rotations Discrepancy Renormalization Ostrowski expansion

Citation

Bromberg, Michael; Ulcigrai, Corinna. A temporal central limit theorem for real-valued cocycles over rotations. Ann. Inst. H. Poincaré Probab. Statist. 54 (2018), no. 4, 2304--2334. doi:10.1214/17-AIHP872. https://projecteuclid.org/euclid.aihp/1539849800


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