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


Bromberg, Michael; Ulcigrai, Corinna (2018). A temporal central limit theorem for real-valued cocycles over rotations. Annales de l'Institut Henri Poincaré (B) Probabilities et Statistiques, 54(4):2304-2334.

Abstract

We consider deterministic random walks on the real line driven by irrational rotations, or equivalently, skew product extensions of a rotation by α where the skewing cocycle is a piecewise constant mean zero function with a jump by one at a point $β$. When $α$ is badly approximable and $β$ is badly approximable with respect to α, 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 $α$ is quadratic irrational, $β$ 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 $β$, 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.

Abstract

We consider deterministic random walks on the real line driven by irrational rotations, or equivalently, skew product extensions of a rotation by α where the skewing cocycle is a piecewise constant mean zero function with a jump by one at a point $β$. When $α$ is badly approximable and $β$ is badly approximable with respect to α, 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 $α$ is quadratic irrational, $β$ 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 $β$, 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.

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Additional indexing

Item Type:Journal Article, refereed, original work
Communities & Collections:07 Faculty of Science > Institute of Mathematics
Dewey Decimal Classification:510 Mathematics
Uncontrolled Keywords:Statistics, Probability and Uncertainty, Statistics and Probability
Language:English
Date:1 November 2018
Deposited On:23 Jan 2019 16:08
Last Modified:24 Sep 2019 23:58
Publisher:Elsevier
ISSN:0246-0203
OA Status:Green
Free access at:Publisher DOI. An embargo period may apply.
Publisher DOI:https://doi.org/10.1214/17-aihp872

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