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In recent years, bootstrap methods have drawn attention for their ability toapproximate the laws of "max statistics" in high-dimensional problems. Aleading example of such a statistic is the coordinate-wise maximum of a sampleaverage of $n$ random vectors in $\mathbb{R}^p$. Existing results for thisstatistic show that the bootstrap can work when $n\ll p$, and rates ofapproximation (in Kolmogorov distance) have been obtained with only logarithmicdependence in $p$. Nevertheless, one of the challenging aspects of this settingis 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 ispossible when extra model structure is available. Specifically, we show that ifthe coordinate-wise variances of the observations exhibit decay, then a nearly$n^{-1/2}$ rate can be achieved, independent of $p$. Furthermore, a surprisingaspect of this dimension-free rate is that it holds even when the decay is veryweak. Lastly, we provide examples showing how these ideas can be applied toinference problems dealing with functional and multinomial data.

Bootstrapping Max Statistics in High Dimensions: Near-Parametric Rates Under Weak Variance Decay and Application to Functional and Multinomial Data