Central limit theorem for stationary  products oftoral automorphisms

Jean-Pierre Conze; Stéphane Le Borgne; Mikaël Roger

doi:10.3934/dcds.2012.32.1597

Article Contents

2012, Volume 32, Issue 5: 1597-1626. Doi: 10.3934/dcds.2012.32.1597

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Central limit theorem for stationary products of toral automorphisms

1.
IRMAR, UMR CNRS 6625, Université de Rennes I, Campus de Beaulieu, 35042 Rennes Cedex

Received: May 31, 2010

Revised: September 30, 2011

Published: April 2012

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Abstract

Let $(A_n(\omega))$ be a stationary process in ${\mathcal M}_d^*(\mathbb{Z})$. For a Hölder function $f$ on $\mathbb{T}^d$ we consider the sums $\sum_{k=1}^n f(^t\hskip -3pt A_k(\omega) \, ^t\hskip -3pt A_{k-1}(\omega) \cdots ^t\hskip -3pt A_1(\omega) \, x {\rm \ mod \ } 1)$ and prove a Central Limit Theorem for a.e. $\omega$ in different situations in particular for "kicked" stationary processes. We use the method of multiplicative systems of Komlòs and the Multiplicative Ergodic Theorem.

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Mathematics Subject Classification: 60F05, 37A30, 60G10, 37D99.

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