Asymptotic optimality of the SPRT for the detection of Markov signals

The problem of detecting a Markov signal when a variable number of noisy measurements can be taken is here considered. In particular, the signal--observation sequence $\{X_i,Z_i\}_{i\in\mathbb{N}}$ is a hidden Markov model (HMM) and a sequential probability ratio test (SPRT) is used to detect $\{X_i\}_{i\in\mathbb{N}}$. It is known that the SPRT for testing simple hypotheses based on independent and identically distributed (i.i.d.) observations has a number of remarkable properties, the most appealing being the fact that it simultaneously minimizes the expected sample size under both hypotheses. These properties, however, may fail to hold as the observations $\{Z_i\}_{i\in\mathbb{N}}$ are not independent. In this paper sufficient conditions for the validity of these properties are stated. In particular, it is shown that under a set of rather mild conditions the test ends with probability one and its stopping time is almost surely minimized in the class of tests with the same or smaller error probabilities. Furthermore, reinforcing one of such conditions, it is also shown that any moment of the stopping time distribution is first-order asymptotically minimized in the same class of tests.