Asymptotics for high-dimensional covariance matrices and quadratic forms with applications to the trace functional and shrinkage

TL;DR

This paper develops large sample approximations for bilinear forms of high-dimensional covariance matrices, enabling applications in high-dimensional data analysis such as covariance testing, portfolio optimization, and sparse PCA, without restrictions on dimensions or sample size.

Contribution

It provides the first comprehensive asymptotic theory for bilinear forms of high-dimensional covariance matrices with broad applications, including trace functionals and shrinkage estimation.

Findings

Asymptotic distributions are derived for bilinear forms with various weighting vectors.

Results hold without constraints on dimension, number of forms, or sample size.

Distinct asymptotics are identified for different types of shrinkage vectors.

Abstract

We establish large sample approximations for an arbitray number of bilinear forms of the sample variance-covariance matrix of a high-dimensional vector time series using $ \ell_1$-bounded and small $\ell_2$-bounded weighting vectors. Estimation of the asymptotic covariance structure is also discussed. The results hold true without any constraint on the dimension, the number of forms and the sample size or their ratios. Concrete and potential applications are widespread and cover high-dimensional data science problems such as tests for large numbers of covariances, sparse portfolio optimization and projections onto sparse principal components or more general spanning sets as frequently considered, e.g. in classification and dictionary learning. As two specific applications of our results, we study in greater detail the asymptotics of the trace functional and shrinkage estimation of covariance matrices. In shrinkage estimation, it turns out that the asymptotics differs for weighting vectors bounded away from orthogonaliy and nearly orthogonal ones in the sense that their inner product converges to 0.