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

TL;DR

This paper extends Stein's method using concentration inequalities to multivariate normal approximation, providing error bounds for sums of independent and locally dependent random vectors with explicit moment bounds.

Contribution

It generalizes the concentration inequality approach for Stein's method to the multivariate case, deriving new error bounds for sums of independent and locally dependent vectors.

Findings

Error bound of order $k^{1/2}\gamma$ for standardized sums of independent vectors

Fourth moment bound of order $O_k(1/\sqrt{n})$ for locally dependent vectors

Third moment bound of order $O_k(\log n/\sqrt{n})$ for locally dependent vectors

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 $O_k(\log n/\sqrt{n})$.