Stable central limit theorem in total variation distance
type: Item
instance of: scholarly article
title: Stable central limit theorem in total variation distance
author: Xiang Li
author: Lihu Xu
author: Haoran Yang
published in: Journal of Theoretical Probability
publication date: 2024-12-12T00:00:00Z
review text: Let \(X_1,X_2,\ldots\) be independent and identically distributed \(d\)-dimensional random vectors in the domain of normal attraction of a stable distribution with \(\alpha\in(0,2)\), and whose distribution is locally lower bounded by the Lebesgue measure. Under some boundedness or symmetry assumptions on the associated spectral measure, the authors establish rates of convergence of the distribution of (an appropriate normalisation of) \(X_1+\cdots+X_n\) to a stable law in total variation distance. The case in which the \(X_i\) have a symmetric Pareto distribution is used to show optimality of these convergence rates, and in particular that in the case \(\alpha=1\) the optimal rate is \(n^{-1}\).
reviewed by: Fraser Daly
Mathematics Subject Classification ID: 60F05
Mathematics Subject Classification ID: 60E07
zbMATH DE Number: 7957627
zbMATH Keywords: stable central limit theorem
zbMATH Keywords: total variation distance
zbMATH Keywords: optimal convergence rate
zbMATH Keywords: measure decomposition
MaRDI profile type: Publication
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DOI: 10.1007/S10959-024-01385-7
description: scientific article; zbMATH DE number 7957627