Open Access
2013 Central Limit Theorem for $\mathbb{Z}_{+}^d$-actions by toral endomorphisms
Mordechay Levin
Electron. J. Probab. 18: 1-42 (2013). DOI: 10.1214/EJP.v18-1904


In this paper we prove the central limit theorem for the following multisequence $$\sum_{n_1=1}^{N_1} ... \sum_{n_d=1}^{N_d} f(A_1^{n_1}...A_d^{n_d} {\bf x} )$$ where $f$ is a Hölder's continue function, $A_1,\ldots,A_d$ are $s\times s$ partially hyperbolic commuting integer matrices, and $\bf x$ is a uniformly distributed random variable in $[0,1]^s$. Next we prove the functional central limit theorem, and the almost sure central limit theorem. The main tool is the $S$-unit theorem.


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Mordechay Levin. "Central Limit Theorem for $\mathbb{Z}_{+}^d$-actions by toral endomorphisms." Electron. J. Probab. 18 1 - 42, 2013.


Accepted: 11 March 2013; Published: 2013
First available in Project Euclid: 4 June 2016

zbMATH: 1306.60010
MathSciNet: MR3035763
Digital Object Identifier: 10.1214/EJP.v18-1904

Primary: 60F15
Secondary: 37A

Keywords: central limit theorem , partially hyperbolic actions , toral endomorphisms

Vol.18 • 2013
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