Probabilities of large deviations for the sums of functions of mixing sequences

Let $a_1,a_2,\dots$ be a strictly stationary sequence of random variables satisfying Rosenblatt's mixing condition with coefficient $\alpha(k)\le Ae^{-\alpha k}$, $a,A>0$. We investigate, the probabilites of large deviations (of the order $o(n^{1/8}\ln^{-1}n)$) for the sums
$$ n^{-1/2}(\xi_{1s}+\dots+\xi_{ns}),\qquad\xi_{ks}=f_s(a_k,\dots,a_{k+s-1}),\qquad k=1,2,\dots, $$
where $s=s(n)$, $1\le s(n)\le\ln n$, $|\xi_{1s}|\le B<\infty$, $\mathbf E\xi_{1s}=0$,
$$ \lim_{n\to\infty}n^{-1}(\xi_{1s}+\dots+\xi_{ns})^2>0. $$