# New error bounds in multivariate normal approximations via exchangeable pairs with applications to Wishart matrices and fourth moment theorems

@article{Fang2022NewEB,
title={New error bounds in multivariate normal approximations via exchangeable pairs with applications to Wishart matrices and fourth moment theorems},
author={Xiao Fang and Yuta Koike},
journal={The Annals of Applied Probability},
year={2022}
}
• Published 5 April 2020
• Mathematics
• The Annals of Applied Probability
We extend Stein's celebrated Wasserstein bound for normal approximation via exchangeable pairs to the multi-dimensional setting. As an intermediate step, we exploit the symmetry of exchangeable pairs to obtain an error bound for smooth test functions. We also obtain a continuous version of the multi-dimensional Wasserstein bound in terms of fourth moments. We apply the main results to multivariate normal approximations to Wishart matrices of size $n$ and degree $d$, where we obtain the optimal…
12 Citations

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## References

SHOWING 1-10 OF 50 REFERENCES

### Rates of Convergence in Normal Approximation Under Moment Conditions Via New Bounds on Solutions of the Stein Equation

New bounds for the $$k$$kth-order derivatives of the solutions of the normal and multivariate normal Stein equations are obtained. Our general order bounds involve fewer derivatives of the test

### Stein’s method for normal approximation in Wasserstein distances with application to the multivariate central limit theorem

• Thomas Bonis
• Mathematics, Computer Science
Probability Theory and Related Fields
• 2020
The Stein's method is used to bound the Wasserstein distance of order $2$ between a measure $\nu$ and the Gaussian measure using a stochastic process and is provided to provide optimal convergence rates for the multi-dimensional Central Limit Theorem.

### MULTIVARIATE NORMAL APPROXIMATION USING EXCHANGEABLE PAIRS

• Mathematics
• 2007
Since the introduction of Stein's method in the early 1970s, much research has been done in extending and strengthening it; however, there does not exist a version of Stein's original method of

### Wasserstein distance error bounds for the multivariate normal approximation of the maximum likelihood estimator

• Mathematics, Computer Science
Electronic Journal of Statistics
• 2021
The general bounds are given for possibly high-dimensional, independent and identically distributed random vectors and of the optimal $\mathcal{O}(n^{-1/2})$ order.

### The CLT in high dimensions: Quantitative bounds via martingale embedding

• Computer Science, Mathematics
• 2018
A new method is introduced for obtaining quantitative convergence rates for the central limit theorem (CLT) in a high dimensional setting based on martingale embeddings and specifically on the Skorokhod embedding constructed by the first named author.

### Berry–Esseen bounds of normal and nonnormal approximation for unbounded exchangeable pairs

• Mathematics
The Annals of Probability
• 2019
An exchangeable pair approach is commonly taken in the normal and non-normal approximation using Stein's method. It has been successfully used to identify the limiting distribution and provide an

### A multivariate central limit theorem for Lipschitz and smooth test functions

We provide an abstract multivariate central limit theorem with the Lindeberg type error bounded in terms of Lipschitz functions (Wasserstein 1-distance) or functions with bounded second or third

### Multivariate normal approximation with Stein’s method of exchangeable pairs under a general linearity condition

• Mathematics
• 2009
In this paper we establish a multivariate exchangeable pairs approach within the framework of Stein's method to assess distributional distances to potentially singular multivariate normal

### Two new proofs of the Erdös–Kac Theorem, with bound on the rate of convergence, by Stein's method for distributional approximations

We study the convergence along the central limit theorem for sums of independent tensor powers, $\frac{1}{\sqrt{d}}\sum\limits_{i=1}^d X_i^{\otimes p}$. We focus on the high-dimensional regime where