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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 PublicationProbability Theory and Related Fields
ISSN0178-8051
Volume174Issue: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.

KeywordBismut's Formula Multivariate Approximation Rate Of Convergence Stein’s Method Wasserstein Distance Langevin Algorithm Malliavin Calculus
DOIhttps://doi.org/10.1007/s00440-018-0874-5
Language英语
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Document TypeJournal article
CollectionFaculty of Science and Technology
DEPARTMENT OF MATHEMATICS
Corresponding AuthorLihu Xu
Affiliation1.Department of Statistics, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong
2.Department of Mathematics, Faculty of Science and Technology, University of Macau, Av. Padre Tomás Pereira, Taipa, Macau, China
3.Zhuhai UM Science and Technology Research Institute, Zhuhai, China
Corresponding Author AffilicationFaculty of Science and Technology
Recommended Citation
GB/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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