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

   
@article{EJP1904,
	author = {Mordechay Levin},
	title = {Central Limit Theorem for $\mathbb{Z}_{+}^d$-actions  by toral endomorphisms},
	journal = {Electron. J. Probab.},
	fjournal = {Electronic Journal of Probability},
	volume = {18},
	year = {2013},
	keywords = {Central limit theorem,  partially hyperbolic actions, toral endomorphisms},
	abstract = {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.},
	pages = {no. 35, 1-42},
	issn = {1083-6489},
	doi = {10.1214/EJP.v18-1904},    
        url = {http://ejp.ejpecp.org/article/view/1904}}