  This paper considers a new bootstrap procedure to estimate the distribution
of high-dimensional $\ell_p$-statistics, i.e. the $\ell_p$-norms of the sum of
$n$ independent $d$-dimensional random vectors with $d \gg n$ and $p \in [1,
\infty]$. We provide a non-asymptotic characterization of the sampling
distribution of $\ell_p$-statistics based on Gaussian approximation and show
that the bootstrap procedure is consistent in the Kolmogorov-Smirnov distance
under mild conditions on the covariance structure of the data. As an
application of the general theory we propose a bootstrap hypothesis test for
simultaneous inference on high-dimensional mean vectors. We establish its
asymptotic correctness and consistency under high-dimensional alternatives, and
discuss the power of the test as well as the size of associated confidence
sets. We illustrate the bootstrap and testing procedure numerically on
simulated data.
