THE CLT IN HIGH DIMENSIONS: QUANTITATIVE BOUNDS VIA MARTINGALE EMBEDDING

• 成果类型: Article
• 署名作者: Eldan, Ronen; Mikulincer, Dan; Zhai, Alex
• 署名单位: Weizmann Institute of Science; Stanford University
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/20-AOP1429
• 发表日期: 2020
• 页码: 2494-2524
• 关键词: CENTRAL-LIMIT-THEOREM; berry-esseen bounds; stein kernels; entropy jumps; CONVERGENCE; inequalities; distance; SPACE

摘要

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 Poincare 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 Poincare Probab. Stat. 52 (2016) 1259-1280).