General Asymptotic Bayesian Theory of Quickest Change Detection

The optimal detection procedure for detecting changes in independent and identically distributed (i.i.d.) sequences in a Bayesian setting was derived by Shiryaev in the 1960s. However, the analysis of the performance of this procedure in terms of the average detection delay and false alarm probability has been an open problem. In this paper, we develop a general asymptotic change-point detection theory that is not limited to a restrictive i.i.d. assumption. In particular, we investigate the performance of the Shiryaev procedure for general discrete-time stochastic models in the asymptotic setting, where the false alarm probability approaches zero. We show that the Shiryaev procedure is asymptotically optimal in the general non-i.i.d. case under mild conditions. We also show that the two popular non-Bayesian detection procedures, namely the Page and the Shiryaev–Roberts–Pollak procedures, are generally not optimal (even asymptotically) under the Bayesian criterion. The results of this study are shown to be especially important in studying the asymptotics of decentralized change detection procedures.