A high-dimensional CLT in W2 distance with near optimal convergence rate

• 成果类型: Article
• 署名作者: Zhai, Alex
• 署名单位: Stanford University
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-017-0771-3
• 发表日期: 2018
• 页码: 821-845
• 关键词: CENTRAL-LIMIT-THEOREM; transportation cost; bounds

摘要

Let X-1, ... , X-n be i.i.d. random vectors in R-d with parallel to X-1 parallel to <= beta. Then, we show that 1/root n (X-1 + ... + X-n) converges to a Gaussian in quadratic transportation (also known as Kantorovich or Wasserstein) distance at a rate of O(root d beta log n/root n), improving a result of Valiant and Valiant. The main feature of our theorem is that the rate of convergence is within log n of optimal for n, d -> infinity.