Electronic Journal of Probability

Central Limit Theorem for $\mathbb{Z}_{+}^d$-actions by toral endomorphisms

Mordechay Levin

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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.

Article information

Electron. J. Probab., Volume 18 (2013), paper no. 35, 42 pp.

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60F15: Strong theorems
Secondary: 37A

Central limit theorem partially hyperbolic actions toral endomorphisms

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Levin, Mordechay. Central Limit Theorem for $\mathbb{Z}_{+}^d$-actions by toral endomorphisms. Electron. J. Probab. 18 (2013), paper no. 35, 42 pp. doi:10.1214/EJP.v18-1904. https://projecteuclid.org/euclid.ejp/1465064260

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