Asymptotic optimality of D-CuSum for quickest change detection under transient dynamics

The problem of quickest change detection (QCD) under transient dynamics is studied, in which the change from the initial distribution to the final persistent distribution does not happen instantaneously, but after a series of cascading transient phases. It is assumed that the durations of the transient phases are deterministic but unknown. The goal is to detect the change as quickly as possible subject to a constraint on the average run length to false alarm. The dynamic CuSum (D-CuSum) algorithm is investigated, which is based on reformulating the QCD problem into a dynamic composite hypothesis testing problem, and has a recursion that facilitates implementation. We show that this algorithm is adaptive to the unknown change point, as well as the unknown transient duration. And under mild conditions of the pre-change and post-change distributions, its asymptotic optimality is demonstrated for all possible asymptotic regimes as the transient duration and the average run length to false alarm go to infinity.