We revisit the problem of sequentially testing the mean of bounded distributions in a level-$α$ power-one framework. We study a $\mathrm{KL_{inf}}$-based sequential test that is known to attain the information-theoretic lower bound on the expected stopping time with exact constants as $α\to 0$. Going beyond first-order asymptotics, we establish a central limit theorem (CLT) for the stopping time of this test. Our analysis proceeds in two steps. First, we prove a novel CLT for the $\mathrm{KL_{inf}}$ statistic itself, characterizing its fluctuations around its deterministic limit. We then leverage this result to show that the stopping time, centered appropriately and scaled by $\sqrt{\log(1/α)}$, converges in distribution to a Gaussian limit with an explicit variance. This yields a second-order characterization of an asymptotically optimal sequential test for bounded distributions. Finally, we present numerical experiments that corroborate our theoretical findings.

翻译：我们重新审视了在水平为$α$的势为一框架下对边界分布均值进行顺序检验的问题。我们研究了一种基于$\mathrm{KL_{inf}}$的顺序检验，该检验已知能够在精确常数下达到当$α\to 0$时期望停止时间的信息论下界。超越一阶渐近性，我们建立了该检验停止时间的中心极限定理（CLT）。我们的分析分两步进行。首先，我们证明了$\mathrm{KL_{inf}}$统计量本身的一个新型中心极限定理，刻画了其围绕确定性极限的波动。然后，我们利用这一结果表明，经过恰当中心化并以$\sqrt{\log(1/α)}$尺度化的停止时间，在分布上收敛于具有显式方差的高斯极限。这为边界分布的一个渐近最优顺序检验提供了二阶表征。最后，我们通过数值实验验证了我们的理论发现。