This paper studies the properties of open-ended power-one tests of $H_0: \theta \leqq \theta_0$ versus $H_1: \theta > \theta_0$ or of $H: \theta = \theta_0$ versus $K: \theta \neq \theta_0$ based on sample sums stopped at moving boundaries. The behavior of the expected sample size is analyzed and certain asymptotic results as $\theta \rightarrow \theta_0$ are obtained in the case of a location parameter and also in the case of an exponential family of distributions.
"Power-One Tests Based on Sample Sums." Ann. Statist. 5 (5) 866 - 880, September, 1977. https://doi.org/10.1214/aos/1176343943