Geodesic
On the growth of sums of independent random variables
Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 5, Tome 294 (2002), pp. 158-164

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Some estimates of the growth of sums of independent random variables almost surely are established without any moment conditions. 
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     author = {V. V. Petrov},
     title = {On the growth of sums of independent random variables},
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V. V. Petrov. On the growth of sums of independent random variables. Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 5, Tome 294 (2002), pp. 158-164. http://geodesic.mathdoc.fr/item/ZNSL_2002_294_a11/