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On an asymptotic approach to the change point detection problem and exponential
Teoriâ veroâtnostej i ee primeneniâ, Tome 68 (2023) no. 3, pp. 456-482 Cet article a éte moissonné depuis la source Math-Net.Ru

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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. 
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A. A. Borovkov. On an asymptotic approach to the change point detection problem and exponential. Teoriâ veroâtnostej i ee primeneniâ, Tome 68 (2023) no. 3, pp. 456-482. http://geodesic.mathdoc.fr/item/TVP_2023_68_3_a2/

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