Central limit theorem for endomorphisms of the Euclidean space

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.