  Sequential monitoring of high-dimensional nonlinear time series is studied
for a projection of the second-moment matrix, a problem interesting in its own
right and specifically arising in finance and deep learning. Open-end as well
as closed-end monitoring is studied under mild assumptions on the training
sample and the observations of the monitoring period. Asymptotics is based on
Gaussian approximations of projected partial sums allowing for an estimated
projection vector. Estimation is studied both for classical
non-$\ell_0$-sparsity as well as under sparsity. For the case that the optimal
projection depends on the unknown covariance matrix, hard- and soft-thresholded
estimators are studied. Applications in finance and training of deep neural
networks are discussed. The proposed detectors typically allow to reduce
dramatically the required computational costs as illustrated by monitoring
synthetic data.
