Nearly optimal central limit theorem and bootstrap approximations in high dimensions
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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,…,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)^1/2log 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.
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