Geodesic
Probabilities of~large deviations for sums of~independent random variables with a~common distribution function from the~domain of~attraction of~an~asymmetric stable law
Teoriâ veroâtnostej i ee primeneniâ, Tome 42 (1997) no. 3, pp. 496-530

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Consider a sequence of independent random variables $\{X_i\}$ with a common distribution function $V(x)$ from the domain of attraction of a stable law with an index $\alpha\in(1,2)$ and suppose that $\mathsf{E}X_1=0$ and $$ 0\liminf_{x\to\infty}\frac{1-V(x)}{V(-x)}e^{g(x)}\le\limsup_{x\to\infty}\frac{1-V(x)}{V(-x)}e^{g(x)}\infty, $$ where the positive function $g(x)$ tends to infinity and $$ g(x)x^{-\delta} \text{ increases for} x>x_0 \text{ increases for} \delta1. $$ The paper obtains an asymptotical representation for the probability $\mathsf{P}\{X_1+\dots+X_n>x\}$, which is true uniformly with respect to all positive $x$ for $n$ tending to infinity. The case $\alpha=2$ was earlier carefully investigated in [10].
Keywords: sums of independent random variables, large deviations
Mots-clés : stable distribution, domain of attraction. 
@article{TVP_1997_42_3_a4,
     author = {L. V. Rozovskii},
     title = {Probabilities of~large deviations for sums of~independent random variables with a~common distribution function from the~domain of~attraction of~an~asymmetric stable law},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {496--530},
     publisher = {mathdoc},
     volume = {42},
     number = {3},
     year = {1997},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1997_42_3_a4/}
}
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L. V. Rozovskii. Probabilities of~large deviations for sums of~independent random variables with a~common distribution function from the~domain of~attraction of~an~asymmetric stable law. Teoriâ veroâtnostej i ee primeneniâ, Tome 42 (1997) no. 3, pp. 496-530. http://geodesic.mathdoc.fr/item/TVP_1997_42_3_a4/