Estimates of the concentration functions of weighted sums of independent random variables (Q2874143)

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 \(\Sigma_{k=1}^na_kX_k\) based on the coefficients \(a_k\), \(1 \leq k\leq n\). Results obtained in this paper improve over the recent works in [\textit{O. Friedland} and \textit{S. Sodin}, C. R., Math., Acad. Sci. Paris 345, No. 9, 513--518 (2007; Zbl 1138.60023)] and [\textit{M. Rudelson} and \textit{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)].