Large-Sample Approximations for Variance-Covariance Matrices of High-Dimensional Time Series
(2014) , 22 pages
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- Abstract
- Distributional approximations of (bi-) linear functions of sample variance-covariances matrices play a critical role to analyze vector time series, as they are needed for various purposes, especially to draw inference on the dependence structure in terms of second moments and to analyze projections onto lower dimensional spaces as those generated by principal components. This particularly applies to the high-dimensional case, where the dimension d is allowed to grow with the sample size n and may even be larger than n. We establish large-sample approximations for such bilinear forms related to the sample variance-covariance matrix of a high-dimensional vector time series in terms of strong approximations by Brownian motions. The results cover weakly dependent as well as many long-range dependent linear processes and are valid for uniformly l1-bounded projection vectors, which arise, either naturally or by construction, in many statistical problems extensively studied for high-dimensional series. Among those problems are sparse financial portfolio selection, sparse principal components, shrinkage estimation and change-point analysis for high-dimensional time series, which are discussed in greater detail.
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Steland, A., & von Sachs, R. (2014). Large-Sample Approximations for Variance-Covariance Matrices of High-Dimensional Time Series (ISBA Discussion Paper 2014/31). https://hdl.handle.net/2078.5/197422