  Recent years have seen tremendous advances in the theory and application of
sequential experiments. While these experiments are not always designed with
hypothesis testing in mind, researchers may still be interested in performing
tests after the experiment is completed. The purpose of this paper is to aid in
the development of optimal tests for sequential experiments by analyzing their
asymptotic properties. Our key finding is that the asymptotic power function of
any test can be matched by a test in a limit experiment where a Gaussian
process is observed for each treatment, and inference is made for the drifts of
these processes. This result has important implications, including a powerful
sufficiency result: any candidate test only needs to rely on a fixed set of
statistics, regardless of the type of sequential experiment. These statistics
are the number of times each treatment has been sampled by the end of the
experiment, along with final value of the score (for parametric models) or
efficient influence function (for non-parametric models) process for each
treatment. We then characterize asymptotically optimal tests under various
restrictions such as unbiasedness, \alpha-spending constraints etc. Finally, we
apply our our results to three key classes of sequential experiments: costly
sampling, group sequential trials, and bandit experiments, and show how optimal
inference can be conducted in these scenarios.
