Consistency And Convergence Rate Of Change-point Estimation For Time Series

This paper mainly studies consistency and convergence rate of change-point estimation for time series. Let time series be X₁,…, X_k*, X_k*+1,…, X_n, in whichÏ„* =k*/n is the unknown change point.First, in independent situation, we consider the change point in mean and variance for normal distribution, and present CUSUM estimator of the change point's location, that isÏ„|^. We proved that this estimator is strong consistency with the real location of the change point, and then we gave the convergence rate. Namely, 1. change-point estimation when mean changes(variance doesn't change); 2. change-point estimation when variance changes(mean doesn't change and known); 3. change-point estimation when variance changes(mean doesn't change and unknown). The corresponding convergence rate of change-point estimation is: 0q |Ï„|^ -Ï„* |→0, a.s..Furthermore, if the second moment exits, we used the method of truncations, and proved strong consistency, i.e.Ï„|^-Ï„*→0,a.s.Furthermore, for dependent situation, we consider the change point in mean for NA series, similarly other different dependent circumstances can be proved from this circumstance.