We develop an asymptotic theory for L2 norms of sample mean vectors of
high-dimensional data. An invariance principle for the L2 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 L2 norms. Our
results are applied to multiple tests and inference of covariance matrix
structures.Comment: 3