arXiv subjects
Publications and source records attributed to A. Yu. Zaitsev.
We estimate the concentration functions of $n$-fold convolutions of one-dimensional probability measures. The main result is a supplement to the results of Götze and Zaitsev (1998). We show that the estimation of concentration functions at arguments of bounded size can be reduced to the estimation of these functions at arguments of size $O(\sqrt n)$ which is easier.
A multidimensional version of the results of Komlós, Major and Tusnády for sums of independent random vectors with finite exponential moments is obtained in the particular case where the summands have smooth distributions which are close to Gaussian ones. The bounds obtained reflect this closeness. Furthermore, the results provide sufficient conditions for the existence of i.i.d. vectors $X_1, X_2,\dots$ with given distributions and corresponding i.i.d. Gaussian vectors $Y_1, Y_2,\dots$ such that, for given small $\varepsilon$, $$ {\mathbf P}\Big\{{\limsup\limits_{n\to\infty} \frac1{\log n}\Bigl|\,\sum\limits_{j=1}^n X_j- \sum\limits_{j=1}^n Y_j\,\Bigr|}\le \varepsilon\Big\}=1. $$
The problem of the approximation of convolutions by accompanying laws in the scheme of series satisfying the infinitesimality condition is considered. It is shown that the quality of approximation depends essentially on the choice of centering constants.
Let $X,X_1,...,X_n$ be independent identically distributed random variables. The paper deals with the question about the behavior of the concentration function of the random variable $\sum_{k=1}^{n}a_k X_k$ according to the arithmetic structure of coefficients $a_k$. Recently the interest to this question has increased significantly due to the study of distributions of eigenvalues of random matrices. In this paper we formulate and prove some refinements of the results of Friedland and Sodin (2007) and Rudelson and Vershynin (2009).
In a recent paper the author obtained optimal bounds for the strong Gaussian approximation of sums of independent $\R^d$-valued random vectors with finite exponential moments. The results may be considered as generalizations of well-known results of Komlós--Major--Tusnády and Sakhanenko. The dependence of constants on the dimension $d$ and on distributions of summands is given explicitly. Some related problems are discussed.