Geodesic
Inequalities for the distribution of a~sum of functions of independent random variables
Matematičeskie zametki, Tome 22 (1977) no. 5, pp. 745-758

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Let $\xi=\sum_{i_1,\dots,i_r=1}^nf_{i_1,\dots,i_r=1}(\zeta_{i_1,\dots,i_r=1})$ where $\zeta_1,\dots,\zeta_n$ are independent random variables and the $f_{i_1,\dots,i_r=1}$ are functions (e.g., taking the values 0 and 1). For cases when “almost all” the summands forming $\xi$ are equal to 0 with a probability close to 1, estimates from above and below are obtained for the quantity $\mathsf P\{\xi=0\}$, as well as upper estimates for the distance in variation between the distribution $\xi$, and the distribution of the “approximating” sum of independent random variables. 
@article{MZM_1977_22_5_a12,
     author = {A. M. Zubkov},
     title = {Inequalities for the distribution of a~sum of functions of independent random variables},
     journal = {Matemati\v{c}eskie zametki},
     pages = {745--758},
     publisher = {mathdoc},
     volume = {22},
     number = {5},
     year = {1977},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1977_22_5_a12/}
}
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A. M. Zubkov. Inequalities for the distribution of a~sum of functions of independent random variables. Matematičeskie zametki, Tome 22 (1977) no. 5, pp. 745-758. http://geodesic.mathdoc.fr/item/MZM_1977_22_5_a12/