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Central limit theorem for \(\mathbb{Z}_{+}^d\)-actions by toral endomorphisms. (English) Zbl 1306.60010

Author’s abstract: In this paper, we prove the central limit theorem for the following multisequence \[ \sum_{n_{1}=1}^{N_{1}}\cdots\sum_{n_{d}=1}^{N_{d}}f\left( A_{1}^{n_{1}}\dots A_{d}^{n_{d}}\mathbf{x}\right), \] where \(f\) is a Hölder’s continuous function, \(A_{1},\dots,A_{d}\) are \(s\times s\) partially hyperbolic commuting integer matrices, and \(\mathbf{x}\) is a uniformly distributed random variable in \(\left[ 0,1\right] ^{s}\). Then, we prove the functional central limit theorem, and the almost sure central limit theorem. The main tool is the \({S}\)-unit theorem.

MSC:

60F05 Central limit and other weak theorems
60F17 Functional limit theorems; invariance principles
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