Geodesic
Estimation of the Rate of Convergence in the Multidimensional Central Limit Theorem for Endomorphisms of Euclidean Space
Učënye zapiski Kazanskogo universiteta. Seriâ Fiziko-matematičeskie nauki, Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, Tome 155 (2013) no. 2, pp. 33-43

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Let $W$ be such a nonsingular integer square matrix of order $d$ that $|\mathrm{det}\, W|>1$; $f_i(x)$ are real-valued periodic in each argument Lipschitz-continuous functions defined on the unit hypercube in $R^{\, d}$. We consider $m$-dimensional vectors $(f_1(xW^k),\ldots,f_m(xW^k)), $ $k=1,2,\ldots$ and obtain the estimate of order $O(n^{\varepsilon- 1/2})$ (where $\varepsilon$ is an arbitrarily small number) for the distance between the distribution of the normalized sum of these vectors and the normal distribution at all measurable convex sets from $R^m$.
Keywords: endomorphisms, limit theorem, rate of convergence. 
@article{UZKU_2013_155_2_a2,
     author = {F. G. Gabbasov and V. T. Dubrovin},
     title = {Estimation of the {Rate} of {Convergence} in the {Multidimensional} {Central} {Limit} {Theorem} for {Endomorphisms} of {Euclidean} {Space}},
     journal = {U\v{c}\"enye zapiski Kazanskogo universiteta. Seri\^a Fiziko-matemati\v{c}eskie nauki},
     pages = {33--43},
     publisher = {mathdoc},
     volume = {155},
     number = {2},
     year = {2013},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/UZKU_2013_155_2_a2/}
}
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F. G. Gabbasov; V. T. Dubrovin. Estimation of the Rate of Convergence in the Multidimensional Central Limit Theorem for Endomorphisms of Euclidean Space. Učënye zapiski Kazanskogo universiteta. Seriâ Fiziko-matematičeskie nauki, Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, Tome 155 (2013) no. 2, pp. 33-43. http://geodesic.mathdoc.fr/item/UZKU_2013_155_2_a2/