Change Point Analysis of the Mean

We have seen that under the no change in the mean null hypothesis \(H_0\) , and assuming the observations satisfy a functional version of the central limit theorem (Assumptions 1.1.1 and 1.2.2 ), that the asymptotic distribution of many functionals of the CUSUM process may be computed. Since the CUSUM process arises as the objective function in maximally selecting two sample test statistics to test \(H_0\) versus \(H_A\) , it stands to reason that, in the presence of change points in the series, the functionals of the CUSUM process that we have considered should be consistent in the sense that they diverge in probability to positive infinity as the sample size grows. One goal of this chapter is to carefully quantify the asymptotic behaviour of the CUSUM process in the presence of change points.