Stable central limit theorem in total variation distance

Abstract

Under certain general conditions, we prove that the stable central limit theorem holds in the total variation distance and get its optimal convergence rate for all $\alpha \in (0,2)$. Our method is by two measure decompositions, one step estimates, and a very delicate induction with respect to $\alpha$. One measure decomposition is light tailed and borrowed from \cite{BC16}, while the other one is heavy tailed and indispensable for lifting convergence rate for small $\alpha$. The proof is elementary and composed of the ingredients at the postgraduate level. Our result clarifies that when $\alpha =1$ and $X$ has a symmetric Pareto distribution, the optimal rate is $n^{-1}$ rather than $n^{-1}(\ln n{)}^2$ as conjectured in literatures.