  This article considers change point testing and estimation for a sequence of
high-dimensional data. In the case of testing for a mean shift for
high-dimensional independent data, we propose a new test which is based on
$U$-statistic in Chen and Qin (2010) and utilizes the self-normalization
principle [Shao (2010), Shao and Zhang (2010)]. Our test targets dense
alternatives in the high-dimensional setting and involves no tuning parameters.
To extend to change point testing for high-dimensional time series, we
introduce a trimming parameter and formulate a self-normalized test statistic
with trimming to accommodate the weak temporal dependence. On the theory front,
we derive the limiting distributions of self-normalized test statistics under
both the null and alternatives for both independent and dependent
high-dimensional data. At the core of our asymptotic theory, we obtain weak
convergence of a sequential U-statistic based process for high-dimensional
independent data, and weak convergence of sequential trimmed U-statistic based
processes for high-dimensional linear processes, both of which are of
independent interests. Additionally, we illustrate how our tests can be used in
combination with wild binary segmentation to estimate the number and location
of multiple change points. Numerical simulations demonstrate the
competitiveness of our proposed testing and estimation procedures in comparison
with several existing methods in the literature.
