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Andrei Yu. Zaitsev

Publications and source records attributed to Andrei Yu. Zaitsev.

At least 19 recordsLinked to original sources
arXiv2025

Estimates of transport distance in the central limit theorem

Let $X_1,\ldots,X_n$ be $d$-dimensional independent random vectors bounded with probability one. For simplicity, we assume that they have zero mean values: \begin{equation} \mathbf{P}\{\|X_{j}\|\leτ\}=1,\quad\mathbf{E}\,X_{j}=0,\quad j=1,\ldots, n.\nonumber \end{equation} We study the distribution behavior of the sum $S=X_{1}+\cdots+X_{n}$ as a function of the bounding value $τ$. From the non-uniform Bikelis estimate in the one-dimensional central limit theorem it follows that $$ W_1(F,Φ_σ)\le cτ. $$ with an absolute constant $c$, where $W_1$ is the Kantorovich--Rubinstein--Wasserstein transport distance, $F$ is the distribution of the sum $S$, and $Φ_σ$ is the corresponding normal distribution. The main result of the paper is significantly stronger and more precise. It is claimed that $$ ρ(F,Φ_σ) =\inf\int\exp(|x-y|/cτ)\,dπ(x,y)\le c, $$ where the infimum is taken over all bivariate probability distributions $π$ with marginal distributions $F$ and $Φ_σ$. The result has also been generalized to distributions with sufficiently slowly growing cumulants from the class $\mathcal{A}_{1}(τ)$, introduced in the author's 1986 paper. The possibility of generalizing the result to the multivariate case is discussed.

math.PR↗
arXiv2024

Estimates of stability with respect to the number of summands for distributions of successive sums of independent identically distributed vectors

Let $X_1,\dots, X_n,\dots$ be i.i.d.\ $d$-dimensional random vectors with common distribution $F$. Then $S_n = X_1+\dots+X_n$ has distribution $F^n$ (degree is understood in the sense of convolution). Let $$ ρ_{\mathcal{C}_d}(F,G) = \sup_A |F\{A\} - G\{A\}|, $$ where the supremum is taken over all convex subsets of $\mathbb R^d$. Basic result is as follows. For any nontrivial distribution $F$ there is $c_1(F)$ such that $$ ρ_{\mathcal{C}_d}(F^n, F^{n+1})\leq \frac{c_1(F)}{\sqrt n} $$ for any natural $n$. The distribution $F$ is called trivial if it is concentrated on a hyperplane that does not contain the origin. Clearly, for such $F$ $$ ρ_{\mathcal{C}_d}(F^n, F^{n+1}) = 1. $$ A similar result for the Prokhorov distance is also obtained. For any $d$-dimensional distribution~$F$ there is a $c_2(F)>0$ that depends only on $F$ and such that \begin{multline}\nonumber (F^n)\{A\}\le (F^{n+1})\{A^{c_2(F)}\}+\frac{c_2(F)}{\sqrt{n}} \text{and}\quad (F^{n+1})\{A\}\leq (F^n)\{A^{c_2(F)}\}+\frac{c_2(F)} {\sqrt{n}} \end{multline} for any Borel set $ A $ for all positive integers $n$. Here $A^{\varepsilon }$ is $ \varepsilon $-neighborhood of the set $ A $.

math.PR↗
arXiv2023

Improved applications of Arak's inequalities to the Littlewood-Offord problem

Let $X_1,\ldots,X_n$ be independent identically distributed random variables. In this paper we study the behavior of concentration functions of weighted sums $\sum_{k=1}^{n}X_ka_k $ with respect to the arithmetic structure of coefficients~$a_k$ in the context of the Littlewood--Offord problem. In our recent papers, we discussed the relations between the inverse principles stated by Nguyen, Tao and Vu and similar principles formulated by Arak in his papers from the 1980's. In this paper, we will derive some improved (more general and more precise) consequences of Arak's inequalities applying our new bound in the Littlewood--Offord problem. Moreover, we also obtain an improvement of the estimates used in Rudelson and Vershynin's least common denominator method.

math.PR↗
arXiv2021

Arak Inequalities for Concentration Functions and the Littlewood-Offord Problem

Let $X,X_1,\ldots,X_n$ be independent identically distributed random variables. In this paper we study the behavior of concentration functions of weighted sums $\sum_{k=1}^{n}a_k X_k$ with respect to the arithmetic structure of coefficients $a_k$ in the context of the Littlewood--Offord problem. Concentration results of this type received renewed interest in connection with distributions of singular values of random matrices. Recently, Tao and Vu proposed an Inverse Principle in the Littlewood-Offord problem. We discuss the relations between the Inverse Principle of Tao and Vu as well as that of Nguyen and Vu and a similar principle formulated for sums of arbitrary independent random variables in the work of Arak from the 1980's.

math.PR↗
arXiv2021

Convergence to infinite-dimensional compound Poisson distributions on convex polyhedra

The aim of the present work is to provide a supplement to the authors' paper (2018). It is shown that our results on the approximation of distributions of sums of independent summands by the accompanying compound Poisson laws and the estimates of the proximity of sequential convolutions of multidimensional distributions on convex polyhedra may be almost automatically transferred to the infinite-dimensional case.

math.PR↗
arXiv2018

Estimates for the closeness of convolutions of probability distributions on convex polyhedra

The aim of the present work is to show that the results obtained earlier on the approximation of distributions of sums of independent summands by the accompanying compound Poisson laws and the estimates of the proximity of sequential convolutions of multidimensional distributions may be transferred to the estimation of the closeness of convolutions of probability distributions on convex polyhedra.

math.PR↗
arXiv2018

Moderate deviations for the $L_1$-norm of kernel density estimators

The rate of normal approximation for the integral norm of kernel density estimators is investigated in the case of densities with power-type singularities. The quantities from the formulations of published results by the author are estimated. By assumption, the density tends to zero as a power-type function when the argument tends to infinity. Moreover, the density may have a finite number of power-type zeroes and of points with power-type tending to infinity. For such densities the size of zones of moderate deviations are found.

math.PR↗
arXiv2018

Rare events and Poisson point processes

The aim of the present work is to show that the results obtained earlier on the approximation of distributions of sums of independent terms by the accompanying compound Poisson laws may be interpreted as rather sharp quantitative estimates for the closeness between the sample containing independent observations of rare events and the Poisson point process which is obtained after a Poissonization of the initial sample.

math.PR↗
arXiv2018

New applications of Arak's inequalities to the Littlewood-Offord problem

Let $X,X_1,\ldots,X_n$ be independent identically distributed random variables. In this paper we study the behavior of concentration functions of weighted sums $\sum_{k=1}^{n}X_ka_k $ with respect to the arithmetic structure of coefficients $a_k$ in the context of the Littlewood--Offord problem. In recent papers of Eliseeva, Götze and Zaitsev, we discussed the relations between the inverse principles stated by Nguyen, Tao and Vu and similar principles formulated by Arak in his papers from the 1980's. In this paper, we will derive some more general and more precise consequences of Arak's inequalities providing new results in the context of the Littlewood-Offord problem.

math.PR↗
arXiv2016

Bound for the maximal probability in the Littlewood-Offord problem

The paper deals with studying a connection of the Littlewood--Offord problem with estimating the concentration functions of some symmetric infinitely divisible distributions. It is shown that the values at zero of the concentration functions of weighted sums of i.i.d. random variables may be estimated by the values at zero of the concentration functions of symmetric infinitely divisible distributions with the Lévy spectral measures which are multiples of the sum of delta-measures at $\pm$weights involved in constructing the weighted sums.

math.PR↗
arXiv2015

Arak Inequalities for Concentration Functions and the Littlewood--Offord Problem: a shortened version

Let $X,X_1,\ldots,X_n$ be independent identically distributed random variables. In this paper we study the behavior of concentration functions of weighted sums $\sum_{k=1}^{n} X_k a_k$ with respect to the arithmetic structure of coefficients~$a_k$ in the context of the Littlewood--Offord problem. Concentration results of this type received renewed interest in connection with distributions of singular values of random matrices. Recently, Tao and Vu proposed an Inverse Principle in the Littlewood--Offord problem. We discuss the relations between the Inverse Principle of Tao and Vu as well as that of Nguyen and Vu and a similar principle formulated for sums of arbitrary independent random variables in the work of Arak from the 1980's. This paper is a shortened and edited version of the preprint arXiv:1506.09034. Here we present the results without proofs.

math.PR↗
arXiv2014

Estimates for the concentration functions in the Littlewood--Offord problem

Let $X,X_1,...,X_n$ be independent identically distributed random variables. In this paper we study the behavior of the concentration functions of the weighted sums $\sum\limits_{k=1}^{n}a_k X_k$ with respect to the arithmetic structure of coefficients $a_k$. Such concentration results recently became important in connection with investigations about singular values of random matrices. In this paper we formulate and prove some refinements of a result of Vershynin (R. Vershynin, Invertibility of symmetric random matrices, arXiv:1102.0300. (2011). Published in Random Structures and Algorithms, v. 44, no. 2, 135--182 (2014)).

math.PR↗
arXiv2014

On the Littlewood--Offord problem

The paper deals with studying a connection of the Littlewood--Offord problem with estimating the concentration functions of some symmetric infinitely divisible distributions. Some multivariate generalizations of results of Arak (1980) are given. They show a connection of the concentration function of the sum with the arithmetic structure of supports of distributions of independent random vectors for arbitrary distributions of summands.

math.PR↗
arXiv2014

Explicit rates of approximation in the CLT for quadratic forms

Let $X,X_1,X_2,\ldots$ be i.i.d. ${\mathbb{R}}^d$-valued real random vectors. Assume that ${\mathbf{E}X=0}$, $\operatorname {cov}X=\mathbb{C}$, $\mathbf{E}\Vert X\Vert^2=σ^2$ and that $X$ is not concentrated in a proper subspace of $\mathbb{R}^d$. Let $G$ be a mean zero Gaussian random vector with the same covariance operator as that of $X$. We study the distributions of nondegenerate quadratic forms $\mathbb{Q}[S_N]$ of the normalized sums ${S_N=N^{-1/2}(X_1+\cdots+X_N)}$ and show that, without any additional conditions, \[Δ_N\stackrel{\mathrm{def}}{=}\sup_x\bigl |\mathbf{P}\bigl\{\mathbb{Q}[S_N]\leq x\bigr\}-\mathbf{P}\bigl\{\mathbb{Q}[G]\leq x\bigr\}\bigr|={\mathcal{O}}\bigl(N^{-1}\bigr),\] provided that $d\geq5$ and the fourth moment of $X$ exists. Furthermore, we provide explicit bounds of order ${\mathcal{O}}(N^{-1})$ for $Δ_N$ for the rate of approximation by short asymptotic expansions and for the concentration functions of the random variables $\mathbb{Q}[S_N+a]$, $a\in{\mathbb{R}}^d$. The order of the bound is optimal. It extends previous results of Bentkus and Götze [Probab. Theory Related Fields 109 (1997a) 367-416] (for ${d\ge9}$) to the case $d\ge5$, which is the smallest possible dimension for such a bound. Moreover, we show that, in the finite dimensional case and for isometric $\mathbb{Q}$, the implied constant in ${\mathcal{O}}(N^{-1})$ has the form $c_dσ^d(\det\mathbb{C})^{-1/2}\mathbf {E}\|\mathbb{C}^{-1/2}X\|^4$ with some $c_d$ depending on $d$ only. This answers a long standing question about optimal rates in the central limit theorem for quadratic forms starting with a seminal paper by Esséen [Acta Math. 77 (1945) 1-125].

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