Geodesic
On Evaluating the Concentration Functions
Teoriâ veroâtnostej i ee primeneniâ, Tome 6 (1961) no. 1, pp. 103-105

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Let $\xi_1,\dots,\xi_n$ be independent random variables, $$Q_k\{l\}=\mathop{\sup}\limits_x\mathbf P\{x\leq\xi _k\leq x+l\},\\Q(L)=\mathop{\sup}\limits_x\mathbf P\{{x\leq\xi_1+\cdots+\xi_n\leq x+L}\},\quad s=\sum\limits_{k+1}^n(1-Q_k(l)).$$ Theorem 1. If $L\ge l$, then $$Q(L)\leq\frac{CL}{l\sqrt s},$$ where $C$ is an absolute constant. This is a refinement of the main theorem in [1]. 
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     author = {B. A. Rogozin},
     title = {On {Evaluating} the {Concentration} {Functions}},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
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     year = {1961},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1961_6_1_a8/}
}
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B. A. Rogozin. On Evaluating the Concentration Functions. Teoriâ veroâtnostej i ee primeneniâ, Tome 6 (1961) no. 1, pp. 103-105. http://geodesic.mathdoc.fr/item/TVP_1961_6_1_a8/