Material in Chapter VI of Hajek and Sidak's book is extended to a sequential analysis setting: conditions are given under which a sequence of log-likelihood-ratio processes (log-likelihood-ratios for sequential sampling, represented as jump processes in continuous time) converges weakly to a Wiener process with drift, the drift parameter depending on which hypothesis, in a suitable neighborhood of a null hypothesis, prevails. Conditions for convergence of other "test statistic" processes, related to likelihood ratios, are also given. Asymptotic sequential tests can thereby be constructed. Some "two-sample problem" examples are treated.
"Weak Convergence of Processes Related to Likelihood Ratios." Ann. Statist. 5 (2) 330 - 341, March, 1977. https://doi.org/10.1214/aos/1176343798