Geodesic
Probability inequalities for sums of independent random variables
Teoriâ veroâtnostej i ee primeneniâ, Tome 16 (1971) no. 4, pp. 660-675

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Let $X_1,\dots,X_n$ be independent random variables; $S_n=X_1+\dots+X_n$; $x$, $y_1,\dots,y_n$ be arbitrary positive numbers, $y\ge\max\{y_1,\dots,y_n\}$. Inequalities for large deviations are obtained in the following form $$ \mathbf P(S_n>x)\sum_{i=1}^n\mathbf P(X_i>y_i)+P(x,y,A(t,y)) $$ where $P(\cdot,\cdot,\cdot)$ is some function of three arguments, $A(t,y)$ is the sum of moments of the order $t$ truncated on the level $y$. Applications to the strong law of large numbers are given. 
@article{TVP_1971_16_4_a4,
     author = {D. H. Fuc and S. V. Nagaev},
     title = {Probability inequalities for sums of independent random variables},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {660--675},
     publisher = {mathdoc},
     volume = {16},
     number = {4},
     year = {1971},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1971_16_4_a4/}
}
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D. H. Fuc; S. V. Nagaev. Probability inequalities for sums of independent random variables. Teoriâ veroâtnostej i ee primeneniâ, Tome 16 (1971) no. 4, pp. 660-675. http://geodesic.mathdoc.fr/item/TVP_1971_16_4_a4/