Power one sequential tests exist for weakly compact $\mathscr P$ against $\mathscr P^c$

Abstract

We study power-one sequential testing for an i.i.d. law on a Polish sample space. Given a nonempty composite null class $\Pcal\subseteq\mathcal M_1(\X)$, we ask when there exists a level-$\alpha$ stopping rule that rejects almost surely under every alternative in a prescribed class $\Qcal\subseteq\Pcal^c$. Our main sufficient condition is local weak lower semicontinuity and positivity of the information projection functional \( \Phi_\Pcal(Q):=\inf_{P\in\Pcal}\KL(Q\|P). \) In particular, if $\Pcal$ is weakly compact, then for every $\alpha\in(0,1)$ there is a single level-$\alpha$ sequential test with power one against the entire complement $\Pcal^c$. The proof combines Csisz\'ar's nonasymptotic Sanov bound for weakly closed convex empirical-measure sets with a Lindel\"of countable-subcover argument. We also show that weak lower semicontinuity is sufficient but not necessary by giving examples where discontinuous finite-sample events separate alternatives that weak neighborhoods cannot detect. Finally, we construct an $e$-process that is asymptotically relatively growth-rate optimal under weak compactness. We verify the weak-lower-semicontinuity condition for weakly compact nulls, $f$-divergence balls, several integral probability metric balls, Wasserstein balls on proper spaces, and a number of non-weakly-compact semiparametric examples.