Inequalities for the concentration function

A. L. Mirošnikov   ; B. A. Rogozin

Teoriâ veroâtnostej i ee primeneniâ, Tome 25 (1980) no. 1, pp. 178-183

Résumé

Let $\xi_1,\dots,\xi_n$ be independent random variables, $S_n=\xi_1+\dots+\xi_n$. The concentration function $Q(\xi,\lambda)$ of a random variable $\xi$ is defined by $$ Q(\xi,\lambda)=\sup_x\,\mathbf P\{x\le\xi\le x+\lambda\},\qquad \lambda>0. $$ We prove, that there exists a universal constant $C<\infty$ such that for any $n$ and arbitrary $\lambda_1,\dots,\lambda_n\in(0,2L]$ $$ Q(S_n,L)\le CL\biggl( \sum_{k=1}^n\mathbf{M}\biggl(|\xi_k^s|\wedge\frac{\lambda_k}2\biggr)^2Q^{-2}(\xi_k,\lambda_k)\biggr)^{-1/2}, $$ where $\xi^s_k$ t is the symmetrization of $\xi_k$ and $a\wedge b=\min (a,b)$.