Geodesic
An exponential criterion for the law of the iterated logarithm
Zapiski Nauchnykh Seminarov POMI, Investigations in the theory of probability distributions. Part IV, Tome 85 (1979), pp. 158-168

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New necessary and sufficimt conditions expressed by “exponential moments” of independent summands are found for log log type relations. 
@article{ZNSL_1979_85_a11,
     author = {A. I. Martikainen},
     title = {An exponential criterion for the law of the iterated logarithm},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {158--168},
     publisher = {mathdoc},
     volume = {85},
     year = {1979},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1979_85_a11/}
}
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A. I. Martikainen. An exponential criterion for the law of the iterated logarithm. Zapiski Nauchnykh Seminarov POMI, Investigations in the theory of probability distributions. Part IV, Tome 85 (1979), pp. 158-168. http://geodesic.mathdoc.fr/item/ZNSL_1979_85_a11/