  In this paper, we propose a class of monitoring statistics for a mean shift
in a sequence of high-dimensional observations. Inspired by the recent
U-statistic based retrospective tests developed by Wang et al.(2019) and Zhang
et al.(2020), we advance the U-statistic based approach to the sequential
monitoring problem by developing a new adaptive monitoring procedure that can
detect both dense and sparse changes in real-time. Unlike Wang et al.(2019) and
Zhang et al.(2020), where self-normalization was used in their tests, we
instead introduce a class of estimators for $q$-norm of the covariance matrix
and prove their ratio consistency. To facilitate fast computation, we further
develop recursive algorithms to improve the computational efficiency of the
monitoring procedure. The advantage of the proposed methodology is demonstrated
via simulation studies and real data illustrations.
