  Distributional approximations of (bi--) linear functions of sample
variance-covariance 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 $ \ell_1 $-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, the LASSO,
shrinkage estimation and change-point analysis for high--dimensional time
series, which matter for the analysis of big data and are discussed in greater
detail.
