  This paper is concerned with deriving the limit distributions of stopping
times devised to sequentially uncover structural breaks in the parameters of an
autoregressive moving average, ARMA, time series. The stopping rules are
defined as the first time lag for which detectors, based on CUSUMs and Page's
CUSUMs for residuals, exceed the value of a prescribed threshold function. It
is shown that the limit distributions crucially depend on a drift term induced
by the underlying ARMA parameters. The precise form of the asymptotic is
determined by an interplay between the location of the break point and the size
of the change implied by the drift. The theoretical results are accompanied by
a simulation study and applications to electroencephalography, EEG, and IBM
data. The empirical results indicate a satisfactory behavior in finite samples.
