This article is cited in 11 papers

Short Communications

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

Yu. S. Eliseeva^a, A. Yu. Zaitsev<sup>b

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

Abstract: Let $X_1,\dots,X_n$ 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.

MSC: 60E15

Received: 26.02.2012

DOI: 10.4213/tvp4479

 English version: Theory of Probability and its Applications, 2013, 57:4, 670–678

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