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Estimates for the Concentration Functions in the Littlewood–Offord Problem

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Let X,X1, . . . , Xn be independent, identically distributed random variables. In this paper, we study the behavior of concentration functions of the weighted sums \( {\displaystyle \sum_{k=1}^n{a}_k{X}_k} \) with respect to the arithmetic structure of coefficients ak. Such concentration results recently became important in connection with the study of singular values of random matrices. In this paper, we formulate and prove some refinements of a result of Vershynin (2014).

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Authors and Affiliations

  1. St. Petersburg State University, St. Petersburg, Russia

    Yu. S. Eliseeva

  2. Bielefeld University, Bielefeld, Germany

    F. Götze

  3. St. Petersburg Department of the Steklov Mathematical Institute and St. Petersburg State University, St. Petersburg, Russia

    A. Yu. Zaitsev

Authors
  1. Yu. S. Eliseeva
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  2. F. Götze
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  3. A. Yu. Zaitsev
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Corresponding author

Correspondence to Yu. S. Eliseeva.

Additional information

Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 420, 2013, pp. 50–69.

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Eliseeva, Y.S., Götze, F. & Zaitsev, A.Y. Estimates for the Concentration Functions in the Littlewood–Offord Problem. J Math Sci 206, 146–158 (2015). https://doi.org/10.1007/s10958-015-2299-3

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