$L^2$ Asymptotics for High-Dimensional Data

Abstract

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.

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References in corpus (3)

• A two-sample test for high-dimensional data with applications to gene-set testing
• Test for bandedness of high-dimensional covariance matrices and bandwidth estimation
• A Cramér moderate deviation theorem for Hotelling's $T^2$-statistic with applications to global tests

Cited by in corpus (1)

• A Convex Optimization Approach to High-Dimensional Sparse Quadratic Discriminant Analysis