Cumulative Sum Processes

In this chapter we introduce the basic change point in the mean problem for scalar observations. We see that the most logical and straightforward approaches to detect such a change point lead to the consideration of weighted functionals of the cumulative sum (CUSUM) processes computed from the observed data. As such, we begin by developing a comprehensive asymptotic theory for CUSUM processes under conditions that allow for serial dependence in the observations. This includes a careful analysis of how weights applied to the CUSUM process affect the limiting distribution of its functionals, and extensions to multivariate observations.