  In recent years, bootstrap methods have drawn attention for their ability to
approximate the laws of "max statistics" in high-dimensional problems. A
leading example of such a statistic is the coordinate-wise maximum of a sample
average of $n$ random vectors in $\mathbb{R}^p$. Existing results for this
statistic show that the bootstrap can work when $n\ll p$, and rates of
approximation (in Kolmogorov distance) have been obtained with only logarithmic
dependence in $p$. Nevertheless, one of the challenging aspects of this setting
is that established rates tend to scale like $n^{-1/6}$ as a function of $n$.
  The main purpose of this paper is to demonstrate that improvement in rate is
possible when extra model structure is available. Specifically, we show that if
the coordinate-wise variances of the observations exhibit decay, then a nearly
$n^{-1/2}$ rate can be achieved, independent of $p$. Furthermore, a surprising
aspect of this dimension-free rate is that it holds even when the decay is very
weak. Lastly, we provide examples showing how these ideas can be applied to
inference problems dealing with functional and multinomial data.
