Geodesic
Central limit theorem for endomorphisms of the Euclidean space
Učënye zapiski Kazanskogo universiteta. Seriâ Fiziko-matematičeskie nauki, Kazanskii Gosudarstvennyi Universitet. Uchenye Zapiski. Seriya Fiziko-Matematichaskie Nauki, Tome 148 (2006) no. 2, pp. 54-64

Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

Let $W$ be a non-degenerated integer-valued matrix such that $|\det W|>1$, $f(t)=$ $=~f(t_1,\ldots,t_d)$ be a real function periodic with respect to any argument, $f$ satisfy the condition $|f(t)-f(t')|\le A\|t-t'\|$ where $A$ — $\mathrm{const}$, $t,t'\in\overline\Omega_d=\{t:0\le t_i\le1,\ i=1,\ldots,d\}$. A central limit theorem for the sequence $(f(tW^n))$ with the rest $O(1/n^{1/2-\varepsilon})$ is established where $\varepsilon$ is an arbitrarily small positive number. 
@article{UZKU_2006_148_2_a5,
     author = {V. T. Dubrovin},
     title = {Central limit theorem for endomorphisms of the {Euclidean} space},
     journal = {U\v{c}\"enye zapiski Kazanskogo universiteta. Seri\^a Fiziko-matemati\v{c}eskie nauki},
     pages = {54--64},
     publisher = {mathdoc},
     volume = {148},
     number = {2},
     year = {2006},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/UZKU_2006_148_2_a5/}
}
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V. T. Dubrovin. Central limit theorem for endomorphisms of the Euclidean space. Učënye zapiski Kazanskogo universiteta. Seriâ Fiziko-matematičeskie nauki, Kazanskii Gosudarstvennyi Universitet. Uchenye Zapiski. Seriya Fiziko-Matematichaskie Nauki, Tome 148 (2006) no. 2, pp. 54-64. http://geodesic.mathdoc.fr/item/UZKU_2006_148_2_a5/