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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


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).


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


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

Primary: 60F05 , 60G57

Keywords: central limit theorem , martingale embedding

Rights: Copyright © 2020 Institute of Mathematical Statistics


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Vol.48 • No. 5 • September 2020
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