This article is cited in 3 papers

On asymmetric large deviations problem in the case of the stable limit law

A. V. Nagaev

Taškent

Abstract: 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 $0<\alpha<1$, $\beta>\alpha$ and the functions $l(x)$ and $m(x)$ vary slowly as $x\to\infty$. We study the asymptotic behaviour of
$$ \mathbf P\{\xi_1+\dots+\xi_n<x\}\quad\text{for}\ x=0\ (\inf\{y:\ ny^{-\alpha}l(y)\le 1\}). $$


Received: 08.06.1981

 English version: Theory of Probability and its Applications, 1984, 28:4, 670–680

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