Geodesic
On asymmetric large deviations problem in the case of the stable limit law
Teoriâ veroâtnostej i ee primeneniâ, Tome 28 (1983) no. 4, pp. 637-645

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Let $\xi_j$ be i. i. d. random variables such that for $x\ge x_0$ $$ \mathbf P\{\xi_1>x\}=x^{-\alpha}l(x),\quad\mathbf P\{\xi_1-x\}=x^{-\beta}m(x), $$ where $0\alpha1$, $\beta>\alpha$ and the functions $l(x)$ and $m(x)$ vary slowly as $x\to\infty$. We study the asymptotic behaviour of $$ \mathbf P\{\xi_1+\dots+\xi_n\}\quad\text{for}\ x=0\ (\inf\{y:\ ny^{-\alpha}l(y)\le 1\}). $$ 
@article{TVP_1983_28_4_a1,
     author = {A. V. Nagaev},
     title = {On asymmetric large deviations problem in the case of the stable limit law},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {637--645},
     publisher = {mathdoc},
     volume = {28},
     number = {4},
     year = {1983},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1983_28_4_a1/}
}
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A. V. Nagaev. On asymmetric large deviations problem in the case of the stable limit law. Teoriâ veroâtnostej i ee primeneniâ, Tome 28 (1983) no. 4, pp. 637-645. http://geodesic.mathdoc.fr/item/TVP_1983_28_4_a1/