Multivariate approximations in Wasserstein distance by Steins method and Bismuts formula

Steins 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 Bismuts formula in Malliavin calculus to control the derivatives of the Stein equation solutions by the first derivative of the test function. Combined with Steins 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.