Geodesic
Almost sure limit theorem for sums of random vectors
Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 3, Tome 260 (1999), pp. 186-201 Cet article a éte moissonné depuis la source Math-Net.Ru

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We prove a sufficient condition for almost sure limit theorem for sums of independent random vectors under minimal moment assumptions and extremely mild assumptions about normalizing sequences. We also provide an example which shows that our sufficient condition is nearly optimal, as well as another sufficient condition due to I. Berkes and H. Dehling. 
@article{ZNSL_1999_260_a11,
     author = {M. A. Lifshits},
     title = {Almost sure limit theorem for sums of random vectors},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {186--201},
     year = {1999},
     volume = {260},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1999_260_a11/}
}
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M. A. Lifshits. Almost sure limit theorem for sums of random vectors. Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 3, Tome 260 (1999), pp. 186-201. http://geodesic.mathdoc.fr/item/ZNSL_1999_260_a11/