Estimation of the Rate of Convergence in the Multidimensional Central Limit Theorem for Endomorphisms of Euclidean Space

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 $R^m$.