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

Stable central limit theorem in total variation distance