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Teor. Veroyatnost. i Primenen., 2012, Volume 57, Issue 4, Pages 768–777 (Mi tvp4479)  

This article is cited in 9 scientific papers (total in 9 papers)

Short Communications

Estimates of the concentration functions of weighted sums of independent random variables

Yu. S. Eliseevaa, A. Yu. Zaitsevb

a Saint-Petersburg State University
b St. Petersburg Department of V. A. Steklov Institute of Mathematics, Russian Academy of Sciences

Abstract: Let $X_1,…,X_n$ be independent and identially distributed random variables. The paper deals with obtaining upper bounds on the concentration function of the weighted sums $\sum_{k=1}^na_kX_k$ based on the coefficients $a_k$, $1\leqslant k\leqslant n$. Results obtained in this paper improve over the recent works in [O. Friedland and S. Sodin, C. R., Math., Acad. Sci. Paris 345, No. 9, 513–518 (2007; Zbl 1138.60023)] and [M. Rudelson and R. Vershynin, Adv. Math. 218, No. 2, 600–633 (2008; Zbl 1139.15015), Commun. Pure Appl. Math. 62, No. 12, 1707–1739 (2009; Zbl 1183.15031)].

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

DOI: https://doi.org/10.4213/tvp4479

Full text: PDF file (180 kB)
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English version:
Theory of Probability and its Applications, 2013, 57:4, 670–678

Bibliographic databases:

MSC: 60E15
Received: 26.02.2012

Citation: Yu. S. Eliseeva, A. Yu. Zaitsev, “Estimates of the concentration functions of weighted sums of independent random variables”, Teor. Veroyatnost. i Primenen., 57:4 (2012), 768–777; Theory Probab. Appl., 57:4 (2013), 670–678

Citation in format AMSBIB
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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, “Multivariate estimates for the concentration functions of weighted sums of independent identically distributed random variables”, J. Math. Sci. (N. Y.), 204:1 (2015), 78–89  mathnet  crossref  mathscinet
    2. Yu. S. Eliseeva, F. Götze, A. Yu. Zaitsev, “Estimates for the concentration functions in the Littlewood–Offord problem”, J. Math. Sci. (N. Y.), 206:2 (2015), 146–158  mathnet  crossref
    3. Yu. S. Eliseeva, A. Yu. Zaitsev, “On the Littlewood–Offord problem”, J. Math. Sci. (N. Y.), 214:4 (2016), 467–473  mathnet  crossref  mathscinet
    4. 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
    5. F. Götze, Yu. S. Eliseeva, A. Yu. Zaitsev, “Arak’s 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
    6. 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
    7. A. L. Miroshnikov, N. V. Miller, N. I. Popova, Yu. V. Shvets, “O nekotorykh voprosakh integrirovaniya v mnogomernykh prostranstvakh”, Mezhdunar. nauch.-issled. zhurn., 2017, no. 12-5(66), 30–35  mathnet  crossref
    8. 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
    9. Li J., Madiman M., “A Combinatorial Approach to Small Ball Inequalities For Sums and Differences”, Comb. Probab. Comput., 28:1 (2019), 100–129  crossref  mathscinet  isi  scopus
  • Теория вероятностей и ее применения Theory of Probability and its Applications
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