  We develop an asymptotic theory for $L^2$ norms of sample mean vectors of
high-dimensional data. An invariance principle for the $L^2$ norms is derived
under conditions that involve a delicate interplay between the dimension $p$,
the sample size $n$ and the moment condition. Under proper normalization,
central and non-central limit theorems are obtained. To facilitate the related
statistical inference, we propose a plug-in calibration method and a
re-sampling procedure to approximate the distributions of the $L^2$ norms. Our
results are applied to multiple tests and inference of covariance matrix
structures.
