Geodesic
Approximation of convolutions of multidimensional distributions
Zapiski Nauchnykh Seminarov POMI, Problems of the theory of probability distributions. Part IX, Tome 142 (1985), pp. 68-80 
A. Yu. Zaitsev. Approximation of convolutions of multidimensional distributions. Zapiski Nauchnykh Seminarov POMI, Problems of the theory of probability distributions. Part IX, Tome 142 (1985), pp. 68-80. http://geodesic.mathdoc.fr/item/ZNSL_1985_142_a6/
@article{ZNSL_1985_142_a6,
     author = {A. Yu. Zaitsev},
     title = {Approximation of convolutions of multidimensional distributions},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {68--80},
     year = {1985},
     volume = {142},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1985_142_a6/}
}
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One obtains estimates for the accuracy of the approximation of the distributions of sums of independent random vectors, concentrated in a ball of radius $\tau$ within the accuracy of a small probability $p$, with the aid of various approximating distributions in the Lévy–Prokhorov metric.