Geodesic
On the asymptotic behaviour of one-sided large deviation probabilities
Teoriâ veroâtnostej i ee primeneniâ, Tome 26 (1981) no. 2, pp. 369-372

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Let $X_1,X_2,\dots$ be i. i. d. random variables, $$ \mathbf EX_1=0,\qquad F(x)=\mathbf P\{X_1\},\qquad S_n=X_1+\dots+X_n $$ We prove that if for $x\to\infty$ $$ 1-F(x)\thicksim x^{-\alpha}h(x),\qquad \alpha>1, $$ where $h(x)$ is a slowly varying function, then $$ \mathbf P\{S_n\ge x\}\thicksim n(1-F(x))\qquad\text{for}\ n\to\infty\ \text{and}\ \liminf_{n\to\infty}x/n>0. $$ 
@article{TVP_1981_26_2_a9,
     author = {S. V. Nagaev},
     title = {On the asymptotic behaviour of one-sided large deviation probabilities},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {369--372},
     publisher = {mathdoc},
     volume = {26},
     number = {2},
     year = {1981},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1981_26_2_a9/}
}
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S. V. Nagaev. On the asymptotic behaviour of one-sided large deviation probabilities. Teoriâ veroâtnostej i ee primeneniâ, Tome 26 (1981) no. 2, pp. 369-372. http://geodesic.mathdoc.fr/item/TVP_1981_26_2_a9/