Limit theorems for sums of independent random variables defined on a recurrent random walk

Let $\nu_k$ be a recurrent random walk with finite variance on an integer lattice. Let $\{X_i\}$, $\{X_{ij}\}$ $(-\infty<i,j<\infty)$ be sequences of independent random variables, which are independent of $\{\nu_k\}$, and let $b_n(k,i)$ be a non-random positive variables. The paper deals with the asymptotic (as $n\to\infty$) behaviour of the quantities
$$ S_n=\sum_{k=1}^nX_{\nu_k},\qquad\bar S_n=\sum_{k=1}^{\varkappa_n}X_{\nu_k}, $$
where $\varkappa_n$ is the first moment when the random walk leaves the interval $(-a\sqrt n,b\sqrt n)$, $a>0$, $b>0$,
$$ I_n=\sum_{k=1}^nb_n(k,\nu_k)X_{\nu_k}\qquad I_n=\sum_{k=1}^nb_n(k,\nu_k)\sum_{j=1}^kX_{{\nu_k}j}, $$
and some others.