Geodesic
On estimation of the maximal probability for sums of lattice random variables
Teoriâ veroâtnostej i ee primeneniâ, Tome 18 (1973) no. 4, pp. 842-846 Cet article a éte moissonné depuis la source Math-Net.Ru

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This paper deals with the estimation of the maximal probability for sums of independent unimodal symmetric lattice random variable $\xi_k$. The author proves the following inequality $$ \sup_x\mathbf{P}(S_n=x)\le\sqrt{\frac6{\pi}}\frac{p_0}{\sqrt{n(1-p_0^2)}}\bigl(1+\frac{c}{\sqrt{n}}\bigr) $$ where $S_n=\xi_1+\dots+\xi_n, p_0=\sup_x\mathbf{P}(\xi_k-x)$ and $c$ is an absolute constant (one may take $c=2$). 
@article{TVP_1973_18_4_a17,
     author = {N. G. Gamkrelidze},
     title = {On estimation of the maximal probability for sums of lattice random variables},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {842--846},
     year = {1973},
     volume = {18},
     number = {4},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1973_18_4_a17/}
}
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N. G. Gamkrelidze. On estimation of the maximal probability for sums of lattice random variables. Teoriâ veroâtnostej i ee primeneniâ, Tome 18 (1973) no. 4, pp. 842-846. http://geodesic.mathdoc.fr/item/TVP_1973_18_4_a17/