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

STOCHASTIC PROCESSES AND THEIR APPLICATIONS

卷:128

Article

Steland, Ansgar¹  von Sachs, Rainer² 

[1] Rhein Westfal TH Aachen, Inst Stat, Wullnerstr 3, D-52056 Aachen, Germany

[2] Catholic Univ Louvain, Inst Stat Biostat & Sci Actuarielles ISBA, Voie Roman Pays 20, B-1348 Louvain La Neuve, Belgium

关键词: Brownian motion;    Linear process;    Long memory;    Strong approximation;    Quadratic form;    Trace;   

DOI  :  10.1016/j.spa.2017.10.007

来源: Elsevier

【 摘 要 】

We establish large sample approximations for an arbitrary number of bilinear forms of the sample variance-covariance matrix of a high-dimensional vector time series using l(1)-bounded and small l(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. M 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 differ for weighting vectors bounded away from orthogonality and nearly orthogonal ones in the sense that their inner product converges to 0. (C) 2017 Elsevier B.V. All rights reserved.

【 授权许可】

Free