  Change point detection in high dimensional data has found considerable
interest in recent years. Most of the literature either designs methodology for
a retrospective analysis, where the whole sample is already available when the
statistical inference begins, or considers online detection schemes controlling
the average time until a false alarm. This paper takes a different point of
view and develops monitoring schemes for the online scenario, where high
dimensional data arrives successively and the goal is to detect changes as fast
as possible controlling at the same time the probability of a type I error of a
false alarm. We develop a sequential procedure capable of detecting changes in
the mean vector of a successively observed high dimensional time series with
spatial and temporal dependence. The statistical properties of the method are
analyzed in the case where both, thesample size and dimension tend to infinity.
In this scenario, it is shown that the new monitoring scheme has asymptotic
level alpha under the null hypothesis of no change and is consistent under the
alternative of a change in at least one component of the high dimensional mean
vector. The approach is based on a new type of monitoring scheme for
one-dimensional data which turns out to be often more powerful than the usually
used CUSUM and Page-CUSUM methods, and the component-wise statistics are
aggregated by the maximum statistic. For the analysis of the asymptotic
properties of our monitoring scheme we prove that the range of a Brownian
motion on a given interval is in the domain of attraction of the Gumbel
distribution, which is a result of independent interest in extreme value
theory. The finite sample properties of the new methodology are illustrated by
means of a simulation study and in the analysis of a data example.
