Geodesic
On convergence of series of independent random variables
Zapiski Nauchnykh Seminarov POMI, Problems of the theory of probability distributions. Part X, Tome 158 (1987), pp. 158-160

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Sufficient conditions are given for the convergence almost surely of a series composed of independent symmetric random variables. 
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     author = {V. V. Petrov},
     title = {On convergence of series of independent random variables},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {158--160},
     publisher = {mathdoc},
     volume = {158},
     year = {1987},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1987_158_a15/}
}
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V. V. Petrov. On convergence of series of independent random variables. Zapiski Nauchnykh Seminarov POMI, Problems of the theory of probability distributions. Part X, Tome 158 (1987), pp. 158-160. http://geodesic.mathdoc.fr/item/ZNSL_1987_158_a15/