On Extreme Value Asymptotics of Projected Sample Covariances in High Dimensions with Applications in Finance and Convolutional Networks

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.