In 1976, Lai constructed a nontrivial confidence sequence for the mean $\mu$ of a Gaussian distribution with unknown variance $\sigma$. Curiously, he employed both an improper (right Haar) mixture over $\sigma$ and an improper (flat) mixture over $\mu$. Here, we elaborate carefully on the details of his construction, which use generalized nonintegrable martingales and an extended Ville's inequality. While this does yield a sequential t-test, it does not yield an ``e-process'' (due to the nonintegrability of his martingale). In this paper, we develop two new e-processes and confidence sequences for the same setting: one is a test martingale in a reduced filtration, while the other is an e-process in the canonical data filtration. These are respectively obtained by swapping Lai's flat mixture for a Gaussian mixture, and swapping the right Haar mixture over $\sigma$ with the maximum likelihood estimate under the null, as done in universal inference. We also analyze the width of resulting confidence sequences, which have a curious dependence on the error probability $\alpha$. Numerical experiments are provided along the way to compare and contrast the various approaches.

用广义不可积分的鞅和扩展维尔不等式揭示了已有可信区间构建的瑞曲和非积分鞅的构建细节，同时提出了两种新型e过程和可信区间构建方法，分别采用高斯混合替换Lai的曲面混合和在零假设下使用极大似然估计的右Haar混合，同时分析了可信区间的宽度与错误概率的奇妙关系，并通过数值实验进行了比较和对比。

未知方差下的高斯均值的任意时刻有效的 t 检验和置信序列