Improved Central Limit Theorem and bootstrap approximations in high dimensions

TL;DR

This paper introduces improved bounds for Gaussian and bootstrap approximations of the maximum statistic in high dimensions, enhancing accuracy and applicability in econometric analysis.

Contribution

It develops a novel iterative randomized Lindeberg method that significantly improves approximation bounds and broadens bootstrap method applicability in high-dimensional settings.

Findings

New bounds for distributional approximation errors

Enhanced bootstrap method applicability

Improved accuracy over existing bounds

Abstract

This paper deals with the Gaussian and bootstrap approximations to the distribution of the max statistic in high dimensions. This statistic takes the form of the maximum over components of the sum of independent random vectors and its distribution plays a key role in many high-dimensional econometric problems. Using a novel iterative randomized Lindeberg method, the paper derives new bounds for the distributional approximation errors. These new bounds substantially improve upon existing ones and simultaneously allow for a larger class of bootstrap methods.