# 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…

## 12 Citations

### Normal approximation via non-linear exchangeable pairs

- Mathematics, Computer Science
- 2020

A new functional analytic approach to Stein's method of exchangeable pairs that does not require the pair at hand to satisfy any approximate linear regression property is proposed, and in which respect this approach yields fundamentally better bounds than those in the existing literature.

### Normal approximation via non-linear exchangeable pairs

- Mathematics, Computer Science
- 2020

A new functional analytic approach to Stein’s method of exchangeable pairs that does not require the pair at hand to satisfy any approximate linear regression property is proposed and in which respect this approach yields fundamentally better bounds than those in the existing literature.

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It is shown that the convergences stated above also hold in a functional setting, namely as weak convergence in [Formula: see text], and it can be used to prove convergence in expectation of the empirical spectral distributions of the Wishart matrices to the semicircular law.

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New, nearly optimal bounds are derived for the Gaussian approximation to scaled averages of independent high-dimensional centered random vectors over the class of rectangles in the case when the covariance matrix of the scaled average is non-degenerate.

### From $p$-Wasserstein Bounds to Moderate Deviations

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A new method via p -Wasserstein bounds is used to prove Cram´er-type moderate deviations in (multivariate) normal approximations and gives applications to the combinatorial central limit theorem, Wiener chaos, homogeneous sums and local dependence.

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We study the discrepancy between the distribution of a vector-valued functional of i.i.d. random elements and that of a Gaussian vector. Our main contribution is an explicit bound on the convex…

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### Berry–Esseen bounds for Chernoff-type nonstandard asymptotics in isotonic regression

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This paper derives Berry-Esseen bounds for an important class of non-standard asymptotics in nonparametric statistics with Chernoff-type limiting distributions, with a focus on the isotonic…

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