Geodesic
On necessary and sufficient conditions for the law of the iterated logarithm
Teoriâ veroâtnostej i ee primeneniâ, Tome 22 (1977) no. 1, pp. 18-26

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Let $\{X_n;\,n=1,2,\dots\}$ be a sequence of independent not necessarily identically distributed random variables and $\{a_n;\,n=1,2,\dots\}$ be a non-decreasing sequence of positive numbers such that $a_n\to\infty$. We put $\displaystyle S_n=\sum_{j=1}^n X_j$. Necessary and sufficient conditions are found for the relations $\limsup(S_n/a_n)\le 1$ a.s. and $\limsup(S_n/a_n)=1$ a.s. No assumptions about existence of any moments are made. 
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     author = {A. I. Martikaǐnen and V. V. Petrov},
     title = {On necessary and sufficient conditions for the law of the iterated logarithm},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {18--26},
     publisher = {mathdoc},
     volume = {22},
     number = {1},
     year = {1977},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1977_22_1_a1/}
}
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A. I. Martikaǐnen; V. V. Petrov. On necessary and sufficient conditions for the law of the iterated logarithm. Teoriâ veroâtnostej i ee primeneniâ, Tome 22 (1977) no. 1, pp. 18-26. http://geodesic.mathdoc.fr/item/TVP_1977_22_1_a1/