Stable Central Limit Theorem in Total Variation Distance

Xiang Li (), Lihu Xu () and Haoran Yang ()
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Xiang Li: University of Macau
Lihu Xu: University of Macau
Haoran Yang: Peking University

Journal of Theoretical Probability, 2025, vol. 38, issue 1, 1-51

Abstract: Abstract Under certain general conditions, we prove that the stable central limit theorem holds in total variation distance and get its optimal convergence rate for all $$\alpha \in (0,2)$$ α ∈ ( 0 , 2 ) . Our method is by two measure decompositions, one-step estimates, and a very delicate induction with respect to $$\alpha $$ α . One measure decomposition is light tailed and borrowed from Bally (Bernoulli 22:2442–2485, 2016), while the other one is heavy tailed and indispensable for lifting convergence rate for small $$\alpha $$ α . The proof is elementary and composed of ingredients at the postgraduate level. Our result clarifies that when $$\alpha =1$$ α = 1 and X has a symmetric Pareto distribution, the optimal rate is $$n^{-1}$$ n - 1 rather than $$n^{-1} (\ln n)^2$$ n - 1 ( ln n ) 2 as conjectured in the literature.

Keywords: Stable central limit theorem; Total variation distance; Optimal convergence rate; Measure decomposition; Backward induction on $$\alpha $$ α; 60E07; 60F05 (search for similar items in EconPapers)
Date: 2025
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DOI: 10.1007/s10959-024-01385-7

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