Probability Theory and Related Fields

, Volume 174, Issue 3–4, pp 945–979

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

Article

## 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.

## Keywords

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

60F05 60H07

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