Bounded Stopping Times for a Class of Sequential Bayes Tests

Consider the problem of sequentially testing a null hypothesis vs an alternative hypothesis when the risk function is a linear combination of probability of error in the terminal decision and expected sample size (i.e., constant cost per observation.) Assume that the parameter space is the union of null and alternative, the parameter space is convex, the intersection of null and alternative is empty, and the common boundary of the closures of null and alternative is nonempty and compact. Assume further that observations are drawn from a $p$-dimensional exponential family with an open $p$-dimensional parameter space. Sufficient conditions for Bayes tests to have bounded stopping times are given.