Abstract

I will present an asymptotic theory for the $L^2$ 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 perform the related statistical inference, I will introduce re-sampling procedures to approximate the distributions of the $L^2$ norms. The results are applied to mean tests and inference of covariance matrix structures.