## 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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