Delay time in sequential detection of change

We consider a sequential test procedure which detects possible changes in the mean of observations satisfying a weak invariance principle. Our test statistic is based on weighted CUSUMs of the underlying random variables. In this paper, we study the asymptotic behaviour of the delay time if a change has occurred in the sample after a training period of size m in which the observations stay in control. It turns out that in this situation the limiting distribution of the delay time for m-->[infinity] is normal under a suitable standardization provided the change appeared sufficiently soon after m.