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Distributional approximations of (bi--) linear functions of samplevariance-covariance matrices play a critical role to analyze vector timeseries, as they are needed for various purposes, especially to draw inferenceon the dependence structure in terms of second moments and to analyzeprojections onto lower dimensional spaces as those generated by principalcomponents. This particularly applies to the high-dimensional case, where thedimension $d$ is allowed to grow with the sample size $n$ and may even belarger than $n$. We establish large-sample approximations for such bilinearforms related to the sample variance-covariance matrix of a high-dimensionalvector time series in terms of strong approximations by Brownian motions. Theresults cover weakly dependent as well as many long-range dependent linearprocesses and are valid for uniformly $ \ell_1 $-bounded projection vectors,which arise, either naturally or by construction, in many statistical problemsextensively studied for high-dimensional series. Among those problems aresparse financial portfolio selection, sparse principal components, the LASSO,shrinkage estimation and change-point analysis for high--dimensional timeseries, which matter for the analysis of big data and are discussed in greaterdetail.

Large-sample approximations for variance-covariance matrices of high-dimensional time series