Tracking Stopping Times

Let {(Xi, Yi)}i=1 be a sequence of pairs of random variables, and let S be a bounded stopping time with respect to {Xi}i=1. We propose the problem of finding a stopping time T with respect to {Yi}i=1 that optimally tracks S in the sense that T minimizes the average reaction time E(T − S) while keeping the false-alarm probability P(T < S) below a given threshold α. This problem has applications in many different areas. In this paper we present an application related to communication over a channel with noisy feedback.