A complete characterization of sequential testability and change detectability in i.i.d. models

We give a necessary and sufficient condition for the existence of power-one sequential tests in an i.i.d. composite testing problem. A level-(α) test with power one against every alternative exists if and only if the alternatives are separated from the null by a countable family of finite-block events. We provide other equivalent conditions using randomized fixed-sample tests, bounded finite-block scores, e-processes, reduced-filtration test supermartingales, and a countable cover whose finite-block weak-* closed convex hulls are positively separated in total variation. As a bonus, the constructive proof yields tests have pointwise expected sample size (OQ(log(1/α))). Exactly the same conditions also characterize i.i.d. change detectability under optional-horizon average-run-length control: for every (η>0), they are equivalent to an alarm family ((Tγ)γ≥1) satisfying (ProbP^∞(Tγ≤σ)≤ EP^∞σ/γ) for every null law and every stopping time (σ). In fact, when these conditions hold, we can construct a single e-detector such that every null-law average run length lies between (γ) and ((1+η)γ+1), and having robust Lorden delay (OQ(logγ)).