We study a CUSUM (cumulative sums) procedure for detection of changes in the means of weak 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. 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, for one thing, 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. For another thing, we propose to consider change-aligned principal components that allow to reduce further common assumptions on the eigenstructure under the alternative. This approach extends directly to changes that occur at different time points and in different directions by fusing sufficient information on all changes into the first component. The latter findings are illustrated by a few simulations and compared with existing procedures.
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