On a combinatorial limit theorem

We consider the random variable
$$ \eta_n=\sum_{i=1}^n a_i b_{x_i} $$
where $a_1,\dots,a_n, b_1,\dots,b_n$ are sequences of real numbers and
$$ X=\begin{pmatrix} 1 & 2 & \dots & n\\ x_1 & x_2 & \dots & x_n \end{pmatrix} $$
is a random permutation. Hajek found necessary and sufficient conditions for the asymptotic normality of $\eta_n$ when $X$ takes values in the set of all permutations of degree $n$ with equal probabilities. In this paper, we use a new approach to investigation of the asymptotic behaviour of $\eta_n$. This approach enables to prove the asymptotic normality of $\eta_n$ when $X$ takes values in the set of all permutations with a single cycle with equal probabilities. If $X$ takes values in the set of all permutations, our method gives conditions for the asymptotic normality of $\eta_n$ which are very close to Hajek's ones.