Geodesic
On the law of the iterated logarithm without assumptions about the existence of moments
Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 4, Tome 278 (2001), pp. 177-181

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Sufficient conditions are given for the applicability of the law of the interated logarithm to sequences of independent random variables. These conditions do not include assumptions about the existence of any moments. 
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     author = {V. V. Petrov},
     title = {On the law of the iterated logarithm without assumptions about the existence of moments},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {177--181},
     publisher = {mathdoc},
     volume = {278},
     year = {2001},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2001_278_a8/}
}
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V. V. Petrov. On the law of the iterated logarithm without assumptions about the existence of moments. Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 4, Tome 278 (2001), pp. 177-181. http://geodesic.mathdoc.fr/item/ZNSL_2001_278_a8/