On the Asymptotic Approach to the Change-Point Problem and Exponential Convergence Rate in the Ergodic Theorem for Markov Chains

Under the assumption that the change-point time is large, a Poisson approximation for the distribution of the number of false alarms is obtained. We also find upper bounds for the probability of a ``false alarm” on a given time interval. An asymptotic expansion for the mean delay time of the alarm signal relative to the change-point time is obtained. To get this result, we establish the exponential convergence rate in the ergodic theorem for Markov chains with a positive atom; chains of this kind describe the monitoring of control systems. A game-theoretic approach is employed to obtain asymptotically optimal solutions of the change-point problem.