In this paper, we formulate and prove multidimensional generalizations of results obtained previously by the author and A. Yu. Zaitsev. Let X, X 1 , . . . , X n be independent, identically distributed random variables. We study the behavior of the concentration function of the random variable \( {\displaystyle \sum_{k=1}^n{X}_k{a}_k} \) according to the arithmetic structure of the vectors a k . Recently, interest to this problem increased significantly due to study of distributions of eigenvalues of random matrices. In this paper, we formulate and prove some refinements of results of Rudelson–Vershinin and Friedland–Sodin. Bibliography: 29 titles.
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Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 412, 2013, pp. 121–137.
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Eliseeva, Y.S. Multivariate Estimates for the Concentration Functions of Weighted Sums of Independent, Identically Distributed Random Variables. J Math Sci 204, 78–89 (2015). https://doi.org/10.1007/s10958-014-2188-1
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DOI: https://doi.org/10.1007/s10958-014-2188-1