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 
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/
@article{ZNSL_2002_294_a11,
     author = {V. V. Petrov},
     title = {On the growth of sums of independent random variables},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {158--164},
     year = {2002},
     volume = {294},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2002_294_a11/}
}
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Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

Some estimates of the growth of sums of independent random variables almost surely are established without any moment conditions.