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- W4300619095 abstract "Consider the multivariate Stein equation $Delta f - xcdot nabla f = h(x) - E h(Z)$, 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 $alpha$-Holder ($0<alphaleq1$), all derivatives of order $2$ of $f_h$ are $alpha$-Holder {it up to a $log$ factor}; in particular they are $beta$-Holder 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 Raiv{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 $3$. When only a finite moment of order $2+delta$ is assumed ($0<delta<1$), we obtain the optimal rate in $mathcal O(n^{-frac{delta}{2}})$. All constants are explicit and their dependence on the dimension $d$ is studied when $d$ is large." @default.
- W4300619095 created "2022-10-03" @default.
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- W4300619095 date "2018-05-04" @default.
- W4300619095 modified "2023-09-28" @default.
- W4300619095 title "Regularity of solutions of the Stein equation and rates in the multivariate central limit theorem" @default.
- W4300619095 hasPublicationYear "2018" @default.
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