On limit distributions of a statistic

The random variables
$$ \eta_n(t)=\sum_{i\le nt}a_ib_{x_i},\quad0\le t\le1, $$
are considered, where $a_1,\dots,a_n$ and $b_1,\dots,b_n$ are numerical sequences and
$$ X= \begin{pmatrix} 1&2&\hdots&n \\ x_1&x_2&\hdots&x_n \end{pmatrix} $$
is a random permutation either from the class of all permutations or from the class of permutations with one cycle only.
Conditions are obtained for finite dimensional distributions of $\eta_n(t)$ to converge to those of the limit processes.