Arak Inequalities for Concentration Functions and the Littlewood--Offord Problem

Abstract

Let X, X 1 , . . . , X n be independent identically distributed random variables. In this paper we study the behavior of concentration functions of weighted sums n k=1 X k a k depending on the arithmetic structure of coefficients a k . The results obtained for the last ten years for the concentration functions of weighted sums play an important role in the study of singular numbers of random matrices. Recently, Tao and Vu proposed a so-called inverse principle in the Littlewood-Offord problem. We discuss the relations between this Inverse Principle and a similar principle for sums of arbitrarily distributed independent random variables formulated by Arak in the 1980's. be independent identically distributed (i.i.d.) random variables. Let a = (a 1 , . . . , a n ) = 0, where a k = (a k1 , . . . , a kd ) ∈ R d , k = 1, . . . , n. Starting with seminal papers of Littlewood and Offord [24] and Erdös [13], the behavior of the concentration functions of the weighted sums S a = n k=1 X k a k . is studied intensively. In the sequel, let F a denote the distribution of the sum S a . The first results were obtained for the case τ = 0 and d = 1, that is, here the maximal probability max x∈R P{S a = x}. was investigated. For a detailed history of this part of the problem we refer to a recent review of Nguyen and Vu [28].In the last ten years, refined concentration results for the weighted sums S a play an important role in the study of singular values of random matrices (see, for instance, Nguyen and Vu [27], Rudelson and Vershynin [31], [32], Tao and Vu [35], [36] Vershynin [38]). Recently, the authors of the present paper (see [9], [10], and [12]) improved some of concentration bounds of the papers [18], [31], [32], [38]. These results reflect the dependence of the bounds on the arithmetic structure of coefficients a k under various conditions on the vector a ∈ (R d ) n and on the distribution L(X). Several years ago, Tao and Vu [35] (see also [27]) proposed the so-called inverse principle in the Littlewood-Offord problem (see § 2). In the present paper, we discuss the relations between this inverse principle and similar principles formulated by Arak (see [1] and [2]) in his papers from the 1980's. In the one-dimensional case, Arak has found a connection of the concentration function of the sum with the arithmetic structure of supports of distributions of independent random variables for arbitrary distributions of summands.Apparently the authors of the publications mentioned above were not aware of the results from the papers of Arak [1] and [2]. Although Arak himself did not use the concept of "inverse principle" in his works, in essence such a principle was there formulated. It is related to general bounds for concentration functions of distributions of sums of independent one-dimensional random variables. The results were used for the estimation of the rate of approximation of n-fold convolutions of probability distributions by infinitely divisible ones. Later, the methods based on Arak's inverse principle admitted to pro...

New applications of Arak’s inequalities to the Littlewood–Offord problem

“…Finally, we state improved and generalized versions of Theorems 5 and 6 of [9], see Theorems 9-12 of the present paper. Theorem 3.…”

“…Several years ago, Tao and Vu [19] and Nguyen and Vu [14] proposed the so-called 'inverse principles' in the Littlewood-Offord problem (see Section 2). In the papers of Götze, Eliseeva and Zaitsev [8] and [9], we discussed the relations between these inverse principles and similar principles formulated by Arak (see [1]- [3]) in his papers from the 1980's. In the one-dimensional case, Arak has found a connection of the concentration function of the sum with the arithmetic structure of supports of distributions of independent random variables for arbitrary distributions of summands.…”

“…The following Theorem 5 was obtained in [9] with the use of Theorem 2 (see [9, Theorem 3 and Proposition 1]).…”

“…A discussion concerning the comparison of Theorem 5 with the results of Nguyen, Tao and Vu [14,15,19,20] is given in [8] and [9].…”

“…In Theorems 5-7 in [9], we obtained particular cases of our Theorems 9-12, where b n = n and τ = κ, τ j = κ j , j = 1, . .…”