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