Abstract (EN)

Consider the multivariate Stein equation Δf−x⋅∇f=h(x)−Eh(Z), where Z is a standard d-dimensional Gaussian random vector, and let fh be the solution given by Barbour's generator approach. We prove that, when h is α-H\"older (0<α≤1), all derivatives of order 2 of fh are α-H\"older {\it up to a log factor}; in particular they are β-H\"older for all β∈(0,α), hereby improving existing regularity results on the solution of the multivariate Gaussian Stein equation. For α=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 logn/n−−√ in the classical multivariate CLT in 1-Wasserstein distance, as long as the underlying random variables have finite moment of order 3. When only a finite moment of order 2+δ is assumed (0<δ<1), we obtain the optimal rate in O(n−δ2). All constants are explicit and their dependence on the dimension d is studied when d is large.