Geodesic
On a~property of sums of independent random variables
Teoriâ veroâtnostej i ee primeneniâ, Tome 22 (1977) no. 2, pp. 335-346

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Let $\xi_i$, $j=1,2,\dots$ be independent identically distributed random variables with $\mathbf M\xi_1=0$, $\mathbf D\xi_1=1$. Put $P_n(x)=\mathbf P\{\xi_1+\dots+\xi_n\ge x\}$. In the paper, a class of distributions $P_1(x)$ is described having the following property: for $x\ge x_n$, $n\to\infty$ $$ P_n(x)=nP_1(x)(1+o(1)). $$ The dependence of the sequence $\{x_n\}$ on properties of $P_1(x)$ is also analyzed. 
@article{TVP_1977_22_2_a8,
     author = {A. V. Nagaev},
     title = {On a~property of sums of independent random variables},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {335--346},
     publisher = {mathdoc},
     volume = {22},
     number = {2},
     year = {1977},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1977_22_2_a8/}
}
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A. V. Nagaev. On a~property of sums of independent random variables. Teoriâ veroâtnostej i ee primeneniâ, Tome 22 (1977) no. 2, pp. 335-346. http://geodesic.mathdoc.fr/item/TVP_1977_22_2_a8/