Regularity of solutions of the Stein equation and rates in the multivariate central limit theorem
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Consider the multivariate Stein equation Δ f - x·∇ 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 α-Hölder (0<α≤1), all derivatives of order 2 of f_h are α-Hölder up to a factor; in particular they are β-Hölder 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 Raičraivc2004multivariate. As an application, we prove a near-optimal Berry-Esseen bound of the order n/√(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^-δ/2). All constants are explicit and their dependence on the dimension d is studied when d is large.
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