  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.
