Suppose we observe data from a distribution $P$ and we wish to test the composite null hypothesis that $P\in\mathscr P$ against a composite alternative $P\in \mathscr Q\subseteq \mathscr P^c$. Herbert Robbins and coauthors pointed out around 1970 that, while no batch test can have a level $α\in(0,1)$ and power equal to one, sequential tests can be constructed with this fantastic property. Since then, and especially in the last decade, a plethora of sequential tests have been developed for a wide variety of settings. However, the literature has not yet provided a clean and general answer as to when such power-one sequential tests exist. This paper provides a remarkably general sufficient condition (that we also prove is not necessary). Focusing on i.i.d. laws in Polish spaces without any further restriction, we show that there exists a level-$α$ sequential test for any weakly compact $\mathscr P$, that is power-one against $\mathscr P^c$ (or any subset thereof). We show how to aggregate such tests into an $e$-process for $\mathscr P$ that increases to infinity under $\mathscr P^c$. We conclude by building an $e$-process that is asymptotically relatively growth rate optimal against $\mathscr P^c$, an extremely powerful result.

翻译：假设我们观测到来自分布$P$的数据，并希望检验复合零假设$P\in\mathscr P$与复合备择假设$P\in \mathscr Q\subseteq \mathscr P^c$。赫伯特·罗宾斯及其合作者于1970年左右指出，虽然批量检验无法同时达到水平$α\in(0,1)$和势等于一，但序贯检验可以构造出这一理想特性。此后（尤其是近十年间），学界针对多种场景开发了大量序贯检验方法。然而，现有文献尚未就势一序贯检验存在的充要条件给出清晰且普适的答案。本文提出了一个异常宽泛的充分条件（并证明其非必要条件）。聚焦于波洛尼亚空间中的独立同分布律且不作额外限制，我们证明：对任意弱紧致$\mathscr P$，存在水平为$α$的序贯检验，其针对$\mathscr P^c$（或其任意子集）的势为一。我们进一步展示如何将这些检验聚合成$\mathscr P$的$e$过程，该过程在$\mathscr P^c$下趋于无穷。最后，我们构建了一个对$\mathscr P^c$具有渐近相对增长率最优性的$e$过程，这是一项极为强大的结论。