Back to Search
Start Over
Stable central limit theorem in total variation distance
- Publication Year :
- 2023
-
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.
- Subjects :
- Mathematics - Probability
Subjects
Details
- Database :
- arXiv
- Publication Type :
- Report
- Accession number :
- edsarx.2312.04001
- Document Type :
- Working Paper