Geodesic
Necessary convergence conditions for series $\sum a_nS_n$ in the case of identically distributed independent random quantities
Matematičeskie zametki, Tome 20 (1976) no. 4, pp. 529-536.

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Necessary (in some cases also sufficient) conditions are obtained for convergence of the series $\sum a_nS_n$ where $S_n=\sum_1^n\xi_k$, $\xi_k$ are independent random quantities. The cases in which $\xi_k$ are symmetrical or identically distributed quantities are investigated in more detail. 
@article{MZM_1976_20_4_a7,
     author = {V. F. Gaposhkin},
     title = {Necessary convergence conditions for series $\sum a_nS_n$ in the case of identically distributed independent random quantities},
     journal = {Matemati\v{c}eskie zametki},
     pages = {529--536},
     publisher = {mathdoc},
     volume = {20},
     number = {4},
     year = {1976},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1976_20_4_a7/}
}
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V. F. Gaposhkin. Necessary convergence conditions for series $\sum a_nS_n$ in the case of identically distributed independent random quantities. Matematičeskie zametki, Tome 20 (1976) no. 4, pp. 529-536. http://geodesic.mathdoc.fr/item/MZM_1976_20_4_a7/