  We study a CUSUM (cumulative sums) procedure for the detection of changes in
the means of weakly dependent time series within an abstract Hilbert space
framework. We use an empirical projection approach via a principal component
representation of the data, i.e., we work with the eigenelements of the (long
run) covariance operator. This article contributes to the existing theory in
two directions: By means of a recent result of Reimherr (2015) we show, on one
hand, that the commonly assumed "separation of the leading eigenvalues" for
CUSUM procedures can be avoided. This assumption is not a consequence of the
methodology but merely a consequence of the usual proof techniques. On the
other hand, we propose to consider change-aligned principal components that
allow to further reduce common assumptions on the eigenstructure under the
alternative. This approach extends directly to multidirectional changes, i.e.
changes that occur at different time points and in different directions, by
fusing sufficient information on them into the first component. The latter
findings are illustrated by a few simulations and compared with existing
procedures in a functional data framework.
