Local central limit theorem and potential kernel estimates for a class of symmetric heavy-tailted random variables

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 (Greens function) up to order $mathcal{O} left( |x|^{frac{alpha-2}{3}+varepsilon} right)$ for any $varepsilon>0$ small enough, when $alpha in [1,2)$.