Asymptotics of Sequential Composite Hypothesis Testing under Probabilistic Constraints

We consider the sequential composite binary hypothesis testing problem in which one of the hypotheses is governed by a single distribution while the other is governed by a family of distributions whose parameters belong to a known set F. We would like to design a test to decide which hypothesis is in effect. Under the constraints that the probabilities that the length of the test, a stopping time, exceeds n are bounded by a certain threshold epsilon, we obtain certain fundamental limits on the asymptotic behavior of the sequential test as n tends to infinity. Assuming that Gamma is a convex and compact set, we obtain the set of all first-order error exponents for the problem. We also prove a strong converse. Additionally, under the assumption that Gamma is a finite set, we obtain the set of second-order error exponents.