Lindeberg's method for $α$-stable central limit theorems

This paper is concerned with the Lindeberg's method associated with a (possibly) asymmetric $\alpha$-stable distribution $Y$, in dimension one and $\alpha\in(0,2)$. We use Taylor-like expansion and Dynkin's formula of stable process to extend the Lindeberg's method to the stable central limit theorem and compute the error rate, when the entries are in the domain of normal attraction of a stable law of exponent $\alpha$ and a special case that the entry has a density function with Pareto multiplied by a slowly varying function.