We develop methods for inference following sequential experiments by studying the asymptotic properties of tests. We find that the large-sample
power of any test can be matched by a test in a suitable limit-experiment involving Gaussian diffusions. This establishes that a fixed set of statistics are asymptotically sufficient for testing; these are the number of times each treatment has been sampled, and the final value of the score/efficient influence function process for each treatment. We also derive asymptotically optimal tests under various conditions and apply these findings to three types of sequential experiments: costly sampling, group sequential trials and bandit-experiments.