Multivariate approximations in Wasserstein distance by Stein's method and Bismut's formula

• 成果类型: Article
• 署名作者: Fang, Xiao; Shao, Qi-Man; Xu, Lihu
• 署名单位: Chinese University of Hong Kong; University of Macau
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-018-0874-5
• 发表日期: 2019
• 页码: 945-979
• 关键词: exchangeable pairs; invariant-measures; CONVERGENCE; ergodicity; THEOREM; clt

摘要

Stein's method has been widely used for probability approximations. However, in the multi-dimensional setting, most of the results are for multivariate normal approximation or for test functions with bounded second- or higher-order derivatives. For a class of multivariate limiting distributions, we use Bismut's formula in Malliavin calculus to control the derivatives of the Stein equation solutions by the first derivative of the test function. Combined with Stein's exchangeable pair approach, we obtain a general theorem for multivariate approximations with near optimal error bounds on the Wasserstein distance. We apply the theorem to the unadjusted Langevin algorithm.