Necessary conditions for power-one sequential testing

Determine a necessary condition on a composite null class P ⊂ M1(X) under i.i.d. sampling on a Polish space X that guarantees the existence, for some α ∈ (0,1), of a sequential test τ satisfying sup_{P∈P} P^∞(τ < ∞) ≤ α and Q^∞(τ < ∞) = 1 for every alternative distribution Q in the complement P^c.

Background

The paper proves a very general sufficient condition for the existence of power-one sequential tests: if the composite null class P is weakly compact in the space of Borel probability measures on a Polish space, then for every α ∈ (0,1) there exists a sequential level-α test that almost surely rejects under every alternative Q ∈ P^c. This covers arbitrary i.i.d. laws without requiring domination or parametric structure.

The authors further show that weak lower semicontinuity of Φ(R) = inf_{P∈P} KL(R || P) at each alternative is also a sufficient (but not necessary) condition, providing a counterexample where power-one testing is possible despite failure of weak lower semicontinuity. Consequently, identifying a general necessary condition for the existence of power-one sequential tests—potentially involving topologies finer than the weak topology—remains unresolved.