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Teor. Veroyatnost. i Primenen., 2017, Volume 62, Issue 2, Pages 241–266 (Mi tvp5107)  

This article is cited in 2 scientific papers (total in 2 papers)

Arak inequalities for concentration functions and the Littlewood–Offord problem

F. Götzea, Yu. S. Eliseevab, A. Yu. Zaitsevbc

a Bielefeld University, Department of Mathematics
b Saint Petersburg State University
c St.-Petersburg Department of Steklov Mathematical Institute of Russian Academy of Sciences

Abstract: Let $X,X_1,\ldots,X_n$ be independent identically distributed random variables. In this paper we study the behavior of the concentration functions of the weighted sums $\sum_{k=1}^{n}X_ka_k $ depending on the arithmetic structure of the coefficients $a_k$. The results obtained the last 10 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 for 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 1980s.

Keywords: concentration functions, inequalities, the Littlewood–Offord problem, sums of independent random variables.

Funding Agency Grant Number
Russian Foundation for Basic Research 13-01-00256
16-01-00367
Saint Petersburg State University 6.38.672.2013
Ministry of Education and Science of the Russian Federation 11.G34.31.0026
НШ-2504.2014.1
Russian Academy of Sciences - Federal Agency for Scientific Organizations
Universität Bielefeld SFB 701
The paper was supported by the SFB 701 in Bielefeld, by Laboratory of Chebyshev in St. Petersburg State University (grant of the Government of Russian Federation 11.G34.31.0026), grant of St. Petersburg State University 6.38.672.2013, the Russian Foundation for Basic Research (grants 13-01-00256, 16-01-00367), the Ministry of Higher Education and Science of the Russian Federation (grant NSh-2504.2014.1), and by the Program of Fundamental Research of Russian Academy of Sciences “Modern Problems of Fundamental Mathematics.”


DOI: https://doi.org/10.4213/tvp5107

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English version:
Theory of Probability and its Applications, 2018, 62:2, 196–215

Bibliographic databases:

Received: 11.04.2016
Revised: 30.09.2016
Accepted:20.10.2016

Citation: F. Götze, Yu. S. Eliseeva, A. Yu. Zaitsev, “Arak inequalities for concentration functions and the Littlewood–Offord problem”, Teor. Veroyatnost. i Primenen., 62:2 (2017), 241–266; Theory Probab. Appl., 62:2 (2018), 196–215

Citation in format AMSBIB
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\paper Arak inequalities for concentration functions and the Littlewood--Offord problem
\jour Teor. Veroyatnost. i Primenen.
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\pages 196--215
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. M. A. Lifshits, Ya. Yu. Nikitin, V. V. Petrov, A. Yu. Zaitsev, A. A. Zinger, “Toward the history of the Saint Petersburg school of probability and statistics. I. Limit theorems for sums of independent random variables”, Vestn. St Petersb. Univ. Math., 51:2 (2018), 144–163  crossref  crossref  mathscinet  zmath  isi  elib  scopus
    2. F. Goetze, A. Yu. Zaitsev, “New applications of Arak's inequalities to the Littlewood-Offord problem”, Eur. J. Math., 4:2 (2018), 639–663  crossref  mathscinet  zmath  isi  scopus
  • Теория вероятностей и ее применения Theory of Probability and its Applications
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