Geodesic
Inequalities for the concentration function
Teoriâ veroâtnostej i ee primeneniâ, Tome 25 (1980) no. 1, pp. 178-183 
A. L. Mirošnikov; B. A. Rogozin. Inequalities for the concentration function. Teoriâ veroâtnostej i ee primeneniâ, Tome 25 (1980) no. 1, pp. 178-183. http://geodesic.mathdoc.fr/item/TVP_1980_25_1_a19/
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     author = {A. L. Miro\v{s}nikov and B. A. Rogozin},
     title = {Inequalities for the concentration function},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {178--183},
     year = {1980},
     volume = {25},
     number = {1},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1980_25_1_a19/}
}
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Voir la notice de l'article provenant de la source Math-Net.Ru

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 $a\wedge b=\min (a,b)$.