Geodesic
On the smoothing of probabilities for the sums of integer valued random variables
Teoriâ veroâtnostej i ee primeneniâ, Tome 26 (1981) no. 4, pp. 835-841

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Let $\xi_1,\xi_2,\dots$ be integer valued independent random variables and $S_n=\xi_1+\dots+\xi_n$. We introduce the function $$ \sum_m|\mathbf P\{S_n=m\}-\mathbf P\{S_n=m-1\}| $$ as a measure of the «smoothness» of the distribution $S_n$. Some properties of this function are investigated. 
@article{TVP_1981_26_4_a16,
     author = {N. G. Gamkrelidze},
     title = {On the smoothing of probabilities for the sums of integer valued random variables},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {835--841},
     publisher = {mathdoc},
     volume = {26},
     number = {4},
     year = {1981},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1981_26_4_a16/}
}
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N. G. Gamkrelidze. On the smoothing of probabilities for the sums of integer valued random variables. Teoriâ veroâtnostej i ee primeneniâ, Tome 26 (1981) no. 4, pp. 835-841. http://geodesic.mathdoc.fr/item/TVP_1981_26_4_a16/