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 Multivariate approximations in Wasserstein distance by Stein’s method and Bismut’s formula Xiao Fang1; Qi-Man Shao1; Lihu Xu2,3 2019-08 Source Publication Probability Theory and Related Fields ISSN 0178-8051 Volume 174Issue:3-4Pages:945–979 Abstract 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. Keyword Bismut's Formula Multivariate Approximation Rate Of Convergence Stein’s Method Wasserstein Distance Langevin Algorithm Malliavin Calculus DOI https://doi.org/10.1007/s00440-018-0874-5 Language 英语 Fulltext Access Citation statistics Document Type Journal article Collection Faculty of Science and TechnologyDEPARTMENT OF MATHEMATICS Corresponding Author Lihu Xu Affiliation 1.Department of Statistics, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong2.Department of Mathematics, Faculty of Science and Technology, University of Macau, Av. Padre Tomás Pereira, Taipa, Macau, China3.Zhuhai UM Science and Technology Research Institute, Zhuhai, China Corresponding Author Affilication Faculty of Science and Technology Recommended CitationGB/T 7714 Xiao Fang,Qi-Man Shao,Lihu Xu. Multivariate approximations in Wasserstein distance by Stein’s method and Bismut’s formula[J]. Probability Theory and Related Fields,2019,174(3-4):945–979. APA Xiao Fang,Qi-Man Shao,&Lihu Xu.(2019).Multivariate approximations in Wasserstein distance by Stein’s method and Bismut’s formula.Probability Theory and Related Fields,174(3-4),945–979. MLA Xiao Fang,et al."Multivariate approximations in Wasserstein distance by Stein’s method and Bismut’s formula".Probability Theory and Related Fields 174.3-4(2019):945–979.
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