The central limit theorem for the sums of functions of mixing sequences

Let $a_1,a_2,\dots$ be a strictly stationary sequence of random variables, $f(x_1,\dots,x_s)$ be a measurable function and
$$ \xi_{ks}=f(a_k,\dots,a_{k+s-1}),\qquad k=1,2,\dots $$
We prove that the central limit theorem holds for $\xi_{ks}$ with the remainder term $O(n^{2\omega^{-1/8}-1/2})$ if the sequence $\{a_k\}$ satisfies Rosenblatt's mixing condition with coefficient $\alpha(k)\le Ak^{-\omega}$ ($A>0$, $\omega>3996$) and for $s=s(n)$, $1\le s(n)\le \ln^2n$, the random variables $\xi_{ks}$ are uniformly bounded with probability 1 and $\mathbf E\xi_{ks}=0$.