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

Let $\{X_i\}_{i=-\infty}^\infty$, $\{\xi_i\}_{i=1}^{\infty}$ be two independet sequences of i.i.d. random variables. Suppose that $\xi_i$ are integralvalued. The paper deals with asymptotic behavior the variable $W_n=n^{-1/2}\sum_{k=1}^n X_{\nu_k}$ under $n\to\infty$. It is shown that the distribution of the $W_n$ converge to the normal distribution and the rate of convergence has the same order as the classical Berry–Esseen estimate.