Geodesic
Limit theorems for sums of independent random variables defined on non-recurrent random walk
Zapiski Nauchnykh Seminarov POMI, Investigations in the theory of probability distributions. Part IV, Tome 85 (1979), pp. 17-29

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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. 
@article{ZNSL_1979_85_a1,
     author = {A. N. Borodin},
     title = {Limit theorems for sums of independent random variables defined on non-recurrent random walk},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {17--29},
     publisher = {mathdoc},
     volume = {85},
     year = {1979},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1979_85_a1/}
}
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A. N. Borodin. Limit theorems for sums of independent random variables defined on non-recurrent random walk. Zapiski Nauchnykh Seminarov POMI, Investigations in the theory of probability distributions. Part IV, Tome 85 (1979), pp. 17-29. http://geodesic.mathdoc.fr/item/ZNSL_1979_85_a1/