Nearly optimal central limit theorem and bootstrap approximations in high dimensions

In this paper, we derive new, nearly optimal bounds for the Gaussian approximation to scaled averages of n independent high-dimensional centered random vectors X1,…,Xn over class rectangles in case when covariance matrix average is nondegenerate. bounded Xi’s, implied bound Kolmogorov distance between distribution and vector takes form C(Bn2log3d/n)1/2logn, where d dimension Bn a uniform envelope constant on components Xi’s. This sharp terms Bn, (up logn) sample size n. addition, show that similar hold multiplier empirical bootstrap approximations. Moreover, establish allow unbounded formulated solely moments Finally, demonstrate can be further improved some special smooth moment-constrained cases.