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
$\Gamma$
. 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.