Geodesic
On the strong law of large numbers and the law of the iterated logarithm for a sequence of independent random variables
Teoriâ veroâtnostej i ee primeneniâ, Tome 15 (1970) no. 3, pp. 520-527 Cet article a éte moissonné depuis la source Math-Net.Ru

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Let $\{X_n\}$ be a sequence of independent random variables with zero means and finite variances, $\{b_n\}$ be an increasing sequence of positive numbers, $b_n\to\infty$, $X_n=o(b_n)$ a.s. Some new conditions are found which are sufficient for the equality $\sum_{j=1}^nX_j=o(b_n)$ a.s. These conditions are expressed in terms of second moments. New sufficient conditions for the law of the iterated logarithm are also obtained. 
@article{TVP_1970_15_3_a7,
     author = {V. A. Egorov},
     title = {On the strong law of large numbers and the law of the iterated logarithm for a~sequence of independent random variables},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {520--527},
     year = {1970},
     volume = {15},
     number = {3},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1970_15_3_a7/}
}
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V. A. Egorov. On the strong law of large numbers and the law of the iterated logarithm for a sequence of independent random variables. Teoriâ veroâtnostej i ee primeneniâ, Tome 15 (1970) no. 3, pp. 520-527. http://geodesic.mathdoc.fr/item/TVP_1970_15_3_a7/