Regularity of solutions of the Stein equation and rates in the multivariate central limit theorem

Understand

Consider the multivariate Stein equation $\Delta f - x\cdot \nabla f = h(x) - E h(Z)$, where $Z$ is a standard $d$-dimensional Gaussian random vector, and let $f\_h$ be the solution given by Barbour's generator approach.

• We prove that, when $h$ is $\alpha$-H\&#34;older ($0<\alpha\leq1$), all derivatives of order $2$ of $f\_h$ are $\alpha$-H\&#34;older {\it up to a $\log$ factor}; in particular they are $\beta$-H\&#34;older for all $\beta \in (0, \alpha)$, hereby improving existing regularity results on the solution of the multivariate Gaussian Stein equation.
• For $\alpha=1$, the regularity we obtain is optimal, as shown by an example given by Rai\v{c} \cite{raivc2004multivariate}.
• As an application, we prove a near-optimal Berry-Esseen bound of the order $\log n/\sqrt n$ in the classical multivariate CLT in $1$-Wasserstein distance, as long as the underlying random variables have finite moment of order $3$.