Large deviations in the central limit theorem for endomorphisms of Euclidean space

Let $W$ be such a nonsingular integer matrix that $|\operatorname{det}W|>1$; $f$ is a real-valued periodic for every argument Lipschitz-continuous function defined on the unit hypercube from $R^d$. For a sequence $(f(tW^n))$, we prove the central limit theorem with large deviations within the interval $[1;\mathrm o(n^{1/8}/\ln n)]$.