Geodesic
Characteristic functions and compactness for distributions of sums of independent random variables
Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 24, Tome 454 (2016), pp. 292-308

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Sequences of distributions of centered sums of independent random variables are considered within a series scheme without supposing classical conditions of uniform asymptotic negligibility and uniform asymptotic constancy. We obtained necessary and sufficient conditions of relative and stochastic compactness for these sequences in terms of characteristic functions of the summed random variables with using their $\tau$-centers. 
@article{ZNSL_2016_454_a18,
     author = {A. A. Khartov},
     title = {Characteristic functions and compactness for distributions of sums of independent random variables},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {292--308},
     publisher = {mathdoc},
     volume = {454},
     year = {2016},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2016_454_a18/}
}
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A. A. Khartov. Characteristic functions and compactness for distributions of sums of independent random variables. Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 24, Tome 454 (2016), pp. 292-308. http://geodesic.mathdoc.fr/item/ZNSL_2016_454_a18/