Geodesic
Some estimates for sums of dependent random variables which hold almost surely
Zapiski Nauchnykh Seminarov POMI, Problems of the theory of probability distributions. Part 3, Tome 55 (1976), pp. 113-116

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Some estimates for sums of dependent random variables are proved. Theorem 1 contains no assumptions about existence of moments of random variables. Estimates of the growth of sums of random variables from a stationary sequence are given in Theorem 2. 
@article{ZNSL_1976_55_a6,
     author = {V. V. Petrov},
     title = {Some estimates for sums of dependent random variables which hold almost surely},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {113--116},
     publisher = {mathdoc},
     volume = {55},
     year = {1976},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1976_55_a6/}
}
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V. V. Petrov. Some estimates for sums of dependent random variables which hold almost surely. Zapiski Nauchnykh Seminarov POMI, Problems of the theory of probability distributions. Part 3, Tome 55 (1976), pp. 113-116. http://geodesic.mathdoc.fr/item/ZNSL_1976_55_a6/