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.
Highlights
In [F], [K], Fortet and Kac proved the central limit theorem for the sumN −1 n=0 f where q ≥ is an integer, x ∈ [0, 1) and f is1-periodic function
We prove some limit theorems for the sum
In this paper we extend this result to the case of endomorphisms: Theorem 4
Summary
In [F], [K], Fortet and Kac proved the central limit theorem (abbreviated CLT) for the sum. Zd+-actions by endomorphisms of s-torus (this result were announced in [Le1], [Le2]). Let A be an invertible s × s matrix with integer entries. It generates a surjective endomorphism on the s-dimensional torus [0, 1)s which we will denote by the same letter A. The dual endomorphism A∗ : Zs → Zs is given by the transpose matrix A(t). An action A by surjectives endomorphisms A1, ..., Ad of [0, 1)s is called partially hyperbolic if for all (n1, ..., nd) ∈ Zd \ {0} none of the eigenvalues of the matrix An1 1 ...Andd are roots of unity. Let I be the s × s identity matrix, q1, ..., qd ≥ 2 pairwise coprime integers, Ai = qiI, i = 1, ..., d.
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