Geodesic
Some inequalities for the distributions of sums of independent random variables
Teoriâ veroâtnostej i ee primeneniâ, Tome 22 (1977) no. 2, pp. 254-263

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Let $X_i$, $i=\overline{1,n}$ be independent random variables, $$ S_n=\sum_1^nX_i,\ F_i(x)=\mathbf P(X_i),\ \overline{\alpha}_k=\int_0^\infty x^t\,dF_k(x). $$ Upper estimates are given for $\mathbf P(S_n\ge x)$ in terms of the sum $$ \sum_{1\le i_1\le\dots\le i_p\le n}\overline{\alpha}_{i_1}\dots\overline{\alpha}_{i_p}. $$ Upper and lower estimates are obtained for $\mathbf M|S_n|^t$, $t>2$. 
@article{TVP_1977_22_2_a3,
     author = {S. V. Nagaev and I. F. Pinelis},
     title = {Some inequalities for the distributions of sums of independent random variables},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {254--263},
     publisher = {mathdoc},
     volume = {22},
     number = {2},
     year = {1977},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1977_22_2_a3/}
}
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S. V. Nagaev; I. F. Pinelis. Some inequalities for the distributions of sums of independent random variables. Teoriâ veroâtnostej i ee primeneniâ, Tome 22 (1977) no. 2, pp. 254-263. http://geodesic.mathdoc.fr/item/TVP_1977_22_2_a3/