Translator Disclaimer
September 2020 The CLT in high dimensions: Quantitative bounds via martingale embedding
Ronen Eldan, Dan Mikulincer, Alex Zhai
Ann. Probab. 48(5): 2494-2524 (September 2020). DOI: 10.1214/20-AOP1429

Abstract

We introduce a new method for obtaining quantitative convergence rates for the central limit theorem (CLT) in a high-dimensional setting. Using our method, we obtain several new bounds for convergence in transportation distance and entropy, and in particular: (a) We improve the best known bound, obtained by the third named author (Probab. Theory Related Fields 170 (2018) 821–845), for convergence in quadratic Wasserstein transportation distance for bounded random vectors; (b) we derive the first nonasymptotic convergence rate for the entropic CLT in arbitrary dimension, for general log-concave random vectors (this adds to (Ann. Inst. Henri Poincaré Probab. Stat. 55 (2019) 777–790), where a finite Fisher information is assumed); (c) we give an improved bound for convergence in transportation distance under a log-concavity assumption and improvements for both metrics under the assumption of strong log-concavity. Our method is based on martingale embeddings and specifically on the Skorokhod embedding constructed in (Ann. Inst. Henri Poincaré Probab. Stat. 52 (2016) 1259–1280).

Citation

Download Citation

Ronen Eldan. Dan Mikulincer. Alex Zhai. "The CLT in high dimensions: Quantitative bounds via martingale embedding." Ann. Probab. 48 (5) 2494 - 2524, September 2020. https://doi.org/10.1214/20-AOP1429

Information

Received: 1 October 2018; Revised: 1 January 2020; Published: September 2020
First available in Project Euclid: 23 September 2020

MathSciNet: MR4152649
Digital Object Identifier: 10.1214/20-AOP1429

Subjects:
Primary: 60F05, 60G57

Rights: Copyright © 2020 Institute of Mathematical Statistics

JOURNAL ARTICLE
31 PAGES

This article is only available to subscribers.
It is not available for individual sale.
+ SAVE TO MY LIBRARY

SHARE
Vol.48 • No. 5 • September 2020
Back to Top