Geodesic
On the Increase of Dispersion of Sums of Independent Random Variables
Teoriâ veroâtnostej i ee primeneniâ, Tome 6 (1961) no. 1, pp. 106-108 
B. A. Rogozin. On the Increase of Dispersion of Sums of Independent Random Variables. Teoriâ veroâtnostej i ee primeneniâ, Tome 6 (1961) no. 1, pp. 106-108. http://geodesic.mathdoc.fr/item/TVP_1961_6_1_a9/
@article{TVP_1961_6_1_a9,
     author = {B. A. Rogozin},
     title = {On the {Increase} of {Dispersion} of {Sums} of {Independent} {Random} {Variables}},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {106--108},
     year = {1961},
     volume = {6},
     number = {1},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1961_6_1_a9/}
}
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Let $\xi_1,\xi_2,\dots,\xi_n$ be independent random variables, $$Q_k\{l\}=\mathop{\sup}\limits_x\mathbf P\{x\leq\xi _k\leq x+l\},\\Q(L)=\mathop{\sup}\limits_x\mathbf P\{{x\leq\xi_1+\cdots+\xi_n\leq x+L}\},\quad s=\sum\limits_{k+1}^n{(1-Q_k(l_k)})l_k^2$$ Theorem 1. If $L>\max l_2$, then $$Q(L)\leq\frac{CL}{l\sqrt s},$$ where $C$ is an absolute constant. Special cases of this theorem correspond to the results of [1]–[4].