  Maximum-type statistics of certain functions of the sample covariance matrix
of high-dimensional vector time series are studied to statistically confirm or
reject the null hypothesis that a data set has been collected under normal
conditions. The approach generalizes the case of the maximal deviation of the
sample autocovariances function from its assumed values. Within a linear time
series framework it is shown that Gumbel-type extreme value asymptotics holds
true. As applications we discuss long-only mimimal-variance portfolio
optimization and subportfolio analysis with respect to idiosyncratic risks, ETF
index tracking by sparse tracking portfolios, convolutional deep learners for
image analysis and the analysis of array-of-sensors data.
