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

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
$$ \mathbf P\{\xi_1+\dots+\xi_n<x\}\quad\text{for}\ x=0\ (\inf\{y:\ ny^{-\alpha}l(y)\le 1\}). $$