  Heavy-tailed errors impair the accuracy of the least squares estimate, which
can be spoiled by a single grossly outlying observation. As argued in the
seminal work of Peter Huber in 1973 [{\it Ann. Statist.} {\bf 1} (1973)
799--821], robust alternatives to the method of least squares are sorely
needed. To achieve robustness against heavy-tailed sampling distributions, we
revisit the Huber estimator from a new perspective by letting the tuning
parameter involved diverge with the sample size. In this paper, we develop
nonasymptotic concentration results for such an adaptive Huber estimator,
namely, the Huber estimator with the tuning parameter adapted to sample size,
dimension, and the variance of the noise. Specifically, we obtain a
sub-Gaussian-type deviation inequality and a nonasymptotic Bahadur
representation when noise variables only have finite second moments. The
nonasymptotic results further yield two conventional normal approximation
results that are of independent interest, the Berry-Esseen inequality and
Cram\'er-type moderate deviation. As an important application to large-scale
simultaneous inference, we apply these robust normal approximation results to
analyze a dependence-adjusted multiple testing procedure for moderately
heavy-tailed data. It is shown that the robust dependence-adjusted procedure
asymptotically controls the overall false discovery proportion at the nominal
level under mild moment conditions. Thorough numerical results on both
simulated and real datasets are also provided to back up our theory.
