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

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.

Bounds for the normal approximation of the maximum likelihood estimator

“…Our bounds also remove an additional constant that appears in the bounds of [4] and [1], and further comparisons between our bounds are given in Remark 3.4. In obtaining our bounds, we use Stein's method and in particular make use of the very recent advances in the literature on optimal (or near-optimal) order Wasserstein distances bounds for the multivariate normal approximation of sums of independent random vectors; see the recent works [8,12,14,15,16,27,31] for important contributions to this body of research. Our results to some extent complement this literature by giving optimal order Wasserstein distance bounds for multivariate normal approximation in the much more general setting of the MLE under general regularity conditions, which is in general a nonlinear statistic.…”

The Prelimit Generator Comparison Approach of Stein’s Method

“…Most recently, another approach to getting derivative bounds based on Bismut's formula from Malliavin calculus was proposed in Fang et al (2019). The authors required the diffusion coefficient to be constant, and the assumptions imposed on the drift were similar to those in Mackey and Gorham (2016).…”

Donsker’s theorem in Wasserstein-1 distance

“…When X = R, there is no particular difficulty to evaluate the K-R distance when µ is the Gaussian distribution. When, X = R d , it is only recently (see [9,12,15] and references therein) that some improvement of the standard Stein's method has been proposed to get the K-R distance to the Gaussian measure on R d . The bottleneck is the estimate of the Lipschitz modulus of the second order derivative of the solution of the Stein's equation when F is only assumed to be Lipschitz continuous.…”

“…Hence, until the very recent papers [9,15], the strategy was to assume that ∇f is Lipschitz, apply once (4) to compute the first derivative of P t f and then apply (3) to this expression:…”