Approximation to the stable law by Lindeberg principle

TL;DR

This paper introduces a novel approach using the Lindeberg principle to approximate one-dimensional asymmetric alpha-stable distributions in the smooth Wasserstein distance, providing new proofs for the stable CLT.

Contribution

It is the first to prove the general stable CLT via the Lindeberg principle and introduces a new method for alpha in (0,1] beyond Fourier analysis.

Findings

First proof of the stable CLT using the Lindeberg principle.

New method for alpha in (0,1] distributions.

Approximation in the smooth Wasserstein distance.

Abstract

By the Lindeberg principle, we develop in this paper an approximation to one dimensional (possibly) asymmetric $\alpha$-stable distributions with $\alpha \in (0,2)$ in the smooth Wasserstein distance. It is the first time that the general stable central limit theorem is proved by the Lindeberg principle, and that this theorem with $\alpha \in (0,1]$ is proved by a new method other than Fourier analysis. Our main tools are a Taylor-like expansion and a Kolmogorov forward equation.