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In this paper, the values of the absolute constants in the Kolmogorov–Rogozin and Kesten inequalities are refined for the concentration function of the sum of independent random vectors in a Hilbert space. [ABSTRACT FROM AUTHOR]

Let X,X, . . . , X be independent, identically distributed random variables. In this paper, we study the behavior of concentration functions of the weighted sums $$ {\displaystyle \sum_{k=1}^n{a}_k{X}_k} $$ with respect to the arithmetic structure of coefficients a. Such concentration results recently became important in connection with the study of singular values of random matrices. In this paper, we formulate and prove some refinements of a result of Vershynin (2014). [ABSTRACT FROM AUTHOR]

The paper deals with studying a connection between the Littlewood-Offord problem and estimating the concentration functions of some symmetric infinitely divisible distributions. Some multivariate generalizations of Arak's results (1980) are given. They establish a relationship of the concentration function of the sum and arithmetic structure of supports of the distributions of independent random vectors for arbitrary distributions of summands. Bibliography: 21 titles. [ABSTRACT FROM AUTHOR]

In this paper, we formulate and prove multidimensional generalizations of results obtained previously by the author and A. Yu. Zaitsev. Let X, X , . . . , X 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. 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. [ABSTRACT FROM AUTHOR]

The aim of the present work is evaluation of absolute constants in the Arak inequalities for the concentration functions of convolutions of probability distributions. This result allows us to calculate the constant in the inequality for the uniform distance between n and (n + 1)-fold convolutions of one-dimensional symmetric probability distributions with a characteristic function separated from −1, as well as a number of other estimates, in particular, the accuracy of approximation of samples of rare events by the Poisson point process. [ABSTRACT FROM AUTHOR]