Multivariate Normal Approximation by Stein's Method: The Concentration Inequality Approach

Abstract

The concentration inequality approach for normal approximation by Stein's method is generalized to the multivariate setting. We use this approach to prove a non-smooth function distance for multivariate normal approximation for standardized sums of $k$-dimensional independent random vectors $W={\sum }_{i=1}^n{X}_i$ with an error bound of order $k^{1/2}\gamma$ where $\gamma ={\sum }_{i=1}^{n}E|X_i{|}^3$. For sums of locally dependent (unbounded) random vectors, we obtain a fourth moment bound which is typically of order $O_k(1/\sqrt{n})$, as well as a third moment bound which is typically of order $O_k(\log n/\sqrt{n})$.

38 pages

References in corpus (4)

• Normal approximation under local dependence
• Multivariate normal approximation with Stein's method of exchangeable pairs under a general linearity condition
• Multivariate normal approximation using exchangeable pairs
• An Exposition of Götze's Estimation of the Rate of Convergence in the Multivariate Central Limit Theorem

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