What are Stein's bounds for multivariate distributions?

Question

Stein's bound in the total variation distance $d_{TV}$ expresses that if $F$ is an integrable real random variable, and $N$ a random variable with the standard normal distribution $\mathcal {N}(0,1)$, then $$ d_{TV} (F, N) \leq \sup | E(f'(F)) - E (F f(F)) | $$ where the supremum runs over the smooth (absolutely continuous) functions $f : \mathbb{R} \to \mathbb{R}$ such that $\| f\|_\infty \leq \sqrt {\frac \pi 2}$ and $\| f'\|_\infty \leq 2$ (cf. Nourdin-Peccati, 2012, Chapter~3). A similar bound holds for the Wasserstein distance $W_1$ with the class of functions $f$ satisfying $\| f'\|_\infty \leq \sqrt {\frac 2\pi}$.

What would be a version of this bound (in total variation or Wasserstein distance) for $F$ and $N$ random vectors in $\mathbb {R}^d$ ($N$ with the standard normal law with mean zero and covariance matrix the identity)?

Answer 1

See e.g. Stein's method for functions of multivariate normal random variables, in particular, formula (1.3) there: \begin{equation} \nabla^T\Sigma\nabla f(\mathbf{w})-\mathbf{w}^T\nabla f(\mathbf{w})=h(\mathbf{w})-Eh(\Sigma^{1/2}\mathbf{Z}), \end{equation} where $\mathbf{Z}$ denotes a random vector having the standard multivariate normal distribution of dimension $d$. See also further references in that paper.

In the case when each coordinate of a random vector $F=(F_1,\dots,F_d)$ in $\mathbb R^d$ is a certain kind of limit (defined at the bottom of p. 47 of Multivariate normal approximation using Stein's method and Malliavin calculus by Nourdin, Peccati, and Réveillac) of certain smooth cylindrical functionals of an isonormal Gaussian process (indexed by elements of a Hilbert space) , Theorem 3.5 of that paper provides an upper bound on the Wasserstein distance between the distribution of $f$ and a multivariate normal distribution.