An inequality that relates the probability of false alarm of a change-point detection policy to its average run length to detection is proved. By means of this inequality, a lower bound on the rate of detection, when the change occurs after a long delay, is derived.
"A lower bound on the ARL to detection of a change with a probability constraint on false alarm." Ann. Statist. 24 (1) 431 - 435, February 1996. https://doi.org/10.1214/aos/1033066219