Uniform Testability Implies Asymmetric Testability

Abstract

We prove a filtration-level conversion from uniform fixed-time testing to all-sample one-sided error control and uniform future-tail control. Given an adapted deterministic binary test whose worst-case fixed-time error tends to zero, the construction selects update times with prescribed summable error budgets and holds each selected decision until the next update. For every prescribed significance level, the construction yields a test that controls Type I error at every sample size; from each update time onward, the probability of any subsequent error is bounded uniformly over each hypothesis class by the remaining budget. Hence the test is uniformly consistent and eventually correct almost surely under both hypotheses. Applying the conversion to finite-alphabet coordinate processes proves the first implication in Ryabko's Conjecture 5.2.