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Zap. Nauchn. Sem. POMI, 2013, Volume 420, Pages 50–69 (Mi znsl5726)  

This article is cited in 5 scientific papers (total in 5 papers)

Estimates for the concentration functions in the Littlewood–Offord problem

Yu. S. Eliseevaa, F. Götzeb, A. Yu. Zaitsevac

a St. Petersburg State University, St. Petersburg, Russia
b Fakultät für Mathematik, Universität Bielefeld, Postfach 100131, D-33501 Bielefeld, Germany
c St. Petersburg Department of V. A. Steklov Institute of Mathematics of the Russian Academy of Sciences, St. Petersburg, Russia

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 $\sum_{k=1}^na_kX_k$ 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 (2011).

Key words and phrases: concentration functions, inequalities, the Littlewood–Offord problem, sums of independent random variables.

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English version:
Journal of Mathematical Sciences (New York), 2015, 206:2, 146–158

UDC: 519
Received: 29.10.2013

Citation: Yu. S. Eliseeva, F. Götze, A. Yu. Zaitsev, “Estimates for the concentration functions in the Littlewood–Offord problem”, Probability and statistics. Part 20, Zap. Nauchn. Sem. POMI, 420, POMI, St. Petersburg, 2013, 50–69; J. Math. Sci. (N. Y.), 206:2 (2015), 146–158

Citation in format AMSBIB
\Bibitem{EliGotZai13}
\by Yu.~S.~Eliseeva, F.~G\"otze, A.~Yu.~Zaitsev
\paper Estimates for the concentration functions in the Littlewood--Offord problem
\inbook Probability and statistics. Part~20
\serial Zap. Nauchn. Sem. POMI
\yr 2013
\vol 420
\pages 50--69
\publ POMI
\publaddr St.~Petersburg
\mathnet{http://mi.mathnet.ru/znsl5726}
\transl
\jour J. Math. Sci. (N. Y.)
\yr 2015
\vol 206
\issue 2
\pages 146--158
\crossref{https://doi.org/10.1007/s10958-015-2299-3}
\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84953351752}


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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. Yu. S. Eliseeva, A. Yu. Zaitsev, “On the Littlewood–Offord problem”, J. Math. Sci. (N. Y.), 214:4 (2016), 467–473  mathnet  crossref  mathscinet
    2. A. Yu. Zaitsev, “Bound for the maximal probability in the Littlewood–Offord problem”, J. Math. Sci. (N. Y.), 219:5 (2016), 743–746  mathnet  crossref  mathscinet
    3. F. Götze, Yu. S. Eliseeva, A. Yu. Zaitsev, “Araks inequalities for concentration functions and the Littlewood–Offord problem”, Doklady Mathematics, 93:2 (2016), 202–206 , arXiv: 1512.02938  crossref  crossref  mathscinet  zmath  isi  scopus
    4. F. Götze, Yu. S. Eliseeva, A. Yu. Zaitsev, “Arak inequalities for concentration functions and the Littlewood–Offord problem”, Theory Probab. Appl., 62:2 (2018), 196–215  mathnet  crossref  crossref  mathscinet  isi  elib
    5. 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
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