  We derive non-asymptotic confidence regions for the mean of a random vector
whose coordinates have an unknown dependence structure. The random vector is
supposed to be either Gaussian or to have a symmetric bounded distribution, and
we observe $n$ i.i.d copies of it. The confidence regions are built using a
data-dependent threshold based on a weighted bootstrap procedure. We consider
two approaches, the first based on a concentration approach and the second on a
direct boostrapped quantile approach. The first one allows to deal with a very
large class of resampling weights while our results for the second are
restricted to Rademacher weights. However, the second method seems more
accurate in practice. Our results are motivated by multiple testing problems,
and we show on simulations that our procedures are better than the Bonferroni
procedure (union bound) as soon as the observed vector has sufficiently
correlated coordinates.
