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}
}
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… 

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