Nearly optimal central limit theorem and bootstrap approximations in high dimensions

By Victor Chernozhukov, D. Chetverikov, Yuta Koike. The Annals of Applied Probability, 2020

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In this paper, we derive new, nearly optimal bounds for the Gaussian approximation to scaled averages of $n$ independent high-dimensional centered random vectors $X_1,\dots,X_n$ over the class of rectangles in the case when the covariance matrix of the scaled average is non-degenerate. In the case of bounded $X_i$'s, the implied bound for the Kolmogorov distance between the distribution of the scaled average and the Gaussian vector takes the form $$C (B^2_n \log^3 d/n)^{1/2} \log n,$$ where $d$ is the dimension of the vectors and $B_n$ is a uniform envelope constant on components of $X_i$'s. This bound is sharp in terms of $d$ and $B_n$, and is nearly (up to $\log n$) sharp in terms of the sample size $n$. In addition, we show that similar bounds hold for the multiplier and empirical bootstrap approximations. Moreover, we establish bounds that allow for unbounded $X_i$'s, formulated solely in terms of moments of $X_i$'s. Finally, we demonstrate that the bounds can be further improved in some special smooth and zero-skewness cases.

Subfields: Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/c773cea4f5700be0b4c1887aca2fa31583123a20

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Improved central limit theorem and bootstrap approximations in high dimensions (2019, result): “Moreover, building on the important results by Lopes in [34] and Kuchibhotla and Rinaldo in [32], [17]”