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

Abstract

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-\(\alpha\) 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 \(O_Q(\log(1/\alpha))\). Exactly the same conditions also characterize i.i.d.\ change detectability under optional-horizon average-run-length control: for every \(\eta>0\), they are equivalent to an alarm family \((T_\gamma)_{\gamma\ge1}\) satisfying \(\Prob_{P^\infty}(T_\gamma\le\sigma)\le \E_{P^\infty}\sigma/\gamma\) for every null law and every stopping time \(\sigma\). In fact, when these conditions hold, we can construct a single e-detector such that every null-law average run length lies between \(\gamma\) and \((1+\eta)\gamma+1\), and having robust Lorden delay \(O_Q(\log\gamma)\).