Abstract:
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.
Keywords:change-point problem, change-point detection, delay time, number of “false
alarms,” Poisson approximation, Markov chain with a positive atom,
exponential convergence rate, asymptotically optimal solution.