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

Abstract

In this article, we study a class of heavy-tailed random vari-ables onZin the domain of attraction of anα-stable random variableof indexα∈(0,2) satisfying a certain expansion of their characteris-tic function. Our results include sharp convergence rates for the local(stable) central limit theorem of ordern−(1+1α), a detailed expansion ofthe characteristic function of a long-range random walk with transitionprobability proportional to|x|−(1+α)andα∈(0,2) and furthermoredetailed asymptotic estimates of the discrete potential kernel (Green’sfunction) up to orderO(|x|α−23+ε)for anyε >0 small enough, whenα∈[1,2).