  In this article, we study a class of heavy-tailed random variables on
$\mathbb{Z}$ in the domain of attraction of an $\alpha$-stable random variable
of index $\alpha \in (0,2)$ satisfying a certain expansion of their
characteristic function. Our results include sharp convergence rates for the
local (stable) central limit theorem of order $n^{- (1+ \frac{1}{\alpha})}$, a
detailed expansion of the characteristic function of a long-range random walk
with transition probability proportional to $|x|^{-(1+\alpha)}$ and $\alpha \in
(0,2)$ and furthermore detailed asymptotic estimates of the discrete potential
kernel (Green's function) up to order $\mathcal{O} \left(
|x|^{\frac{\alpha-2}{3}+\varepsilon} \right)$ for any $\varepsilon>0$ small
enough, when $\alpha \in [1,2)$.
