Geodesic
Local theorems for sums of independent random vectors
Zapiski Nauchnykh Seminarov POMI, Problems of the theory of probability distributions. Part IV, Tome 72 (1977), pp. 84-91 
A. V. Makshanov. Local theorems for sums of independent random vectors. Zapiski Nauchnykh Seminarov POMI, Problems of the theory of probability distributions. Part IV, Tome 72 (1977), pp. 84-91. http://geodesic.mathdoc.fr/item/ZNSL_1977_72_a4/
@article{ZNSL_1977_72_a4,
     author = {A. V. Makshanov},
     title = {Local theorems for sums of independent random vectors},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {84--91},
     year = {1977},
     volume = {72},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1977_72_a4/}
}
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Two multidimensional analogs of the local theorems in $L^1$ due to Prohorov and Gnedenko, respectively, are derived.