Change Detection Method of Time Series as an Optimal Stopping Problem: Constructive Proof of Optimal Solution Theorem

In the past paper, we have formulated the structural Change Point Detection (CPD) problem in time series data as an optimal stopping one using the concept of DP (Dynamic Programming). And, we have shown the optimal solution theorem and its proof using reduction to absurdity. In this paper, we present the direct and constructive proof of the optimal solution theorem for the problem.