Probability Theory and Related Fields

, Volume 174, Issue 3–4, pp 945–979 | Cite as

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

  • Xiao Fang
  • Qi-Man Shao
  • Lihu XuEmail author


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.


Bismut’s formula Langevin algorithm Malliavin calculus Multivariate approximation Rate of convergence Stein’s method Wasserstein distance 

Mathematics Subject Classification

60F05 60H07 



We thank Michel Ledoux for very helpful discussions. We also thank two anonymous referees for their valuable comments which have improved the manuscript considerably. Fang X. was partially supported by Hong Kong RGC ECS 24301617, a CUHK direct grant and a CUHK start-up grant. Shao Q. M. was partially supported by Hong Kong RGC GRF 14302515 and 14304917. Xu L. was partially supported by Macao S.A.R. (FDCT 038/2017/A1, FDCT 030/2016/A1, FDCT 025/2016/A1), NNSFC 11571390, University of Macau (MYRG 2016-00025-FST, MYRG 2018-00133-FST).


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of StatisticsThe Chinese University of Hong KongShatinHong Kong
  2. 2.Department of Mathematics, Faculty of Science and TechnologyUniversity of MacauTaipa, MacauChina
  3. 3.Zhuhai UM Science and Technology Research InstituteZhuhaiChina

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