Arak inequalities for concentration functions and the Littlewood–Offord problem

F. Götze, Yu S. Eliseeva, A. Yu Zaitsev

Research output: Contribution to journalArticleResearchpeer-review

Abstract

Let X, X1, …, Xn be independent identically distributed random variables. In this paper we study the behavior of the concentration functions of the weighted sums (Formula presented) depending on the arithmetic structure of the coefficients ak. 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.

Translated title of the contributionНеравенства Арака для функций концентрации и проблема Литтлвуда–Оффорда
Original languageEnglish
Pages (from-to)196-215
Number of pages20
JournalTheory of Probability and its Applications
Volume62
Issue number2
Early online date15 May 2018
DOIs
StatePublished - 15 May 2018

Keywords

  • Concentration functions
  • Inequalities
  • Sums of independent random variables
  • The Littlewood–Offord problem

Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

Cite this

Götze, F. ; Eliseeva, Yu S. ; Zaitsev, A. Yu. / Arak inequalities for concentration functions and the Littlewood–Offord problem. In: Theory of Probability and its Applications. 2018 ; Vol. 62, No. 2. pp. 196-215.
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Arak inequalities for concentration functions and the Littlewood–Offord problem. / Götze, F.; Eliseeva, Yu S.; Zaitsev, A. Yu.

In: Theory of Probability and its Applications, Vol. 62, No. 2, 15.05.2018, p. 196-215.

Research output: Contribution to journalArticleResearchpeer-review

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