We study asymptotic anytime-valid confidence sequences for degree-two U-statistics under continuous monitoring. In the nondegenerate case, Hoeffding's projection reduces the problem to a time-uniform central limit theory for the partial sums of the first-order projection, while the canonical remainder is shown to be negligible under mild moment assumptions. A leave-one-out jackknife estimator then yields a fully data-driven procedure, leading to confidence sequences with asymptotic coverage guarantee for the parameter of interest. In the degenerate case, we show that the U-statistic is approximated by a centered quadratic Gaussian-chaos rather than by a simple Gaussian, which poses significant challenges for sequential inference. To address this issue, we novelly develop the Spectrally Allocated Gaussian-chaos Excursion (SAGE) boundary, and then provide plug-in implementations based on truncated spectrum estimation with consistency guarantees. The resulting widths can attain the expected time-uniform optimal rates: $\sqrt{\log\log n/n}$ in the nondegenerate regime and $\log\log n/n$ in the degenerate regime. Several widely used U-statistics are discussed within the proposed framework, and numerical experiments further support the validity of the derived theory.

翻译：我们研究了连续监测下二阶U统计量的渐近任意时有效置信序列。在非退化情形下，Hoeffding投影将问题简化为关于一阶投影部分和的时间一致中心极限理论，而正则化余项在温和矩条件下被证明可忽略。基于留一法Jackknife估计构造的完全数据驱动程序可得到关于感兴趣参数的渐近覆盖保证置信序列。在退化情形下，U统计量近似于中心化二次高斯混沌而非简单高斯过程，这为序贯推断带来重大挑战。针对此问题，我们创新性地提出谱分配高斯混沌偏移（SAGE）边界，并基于截断谱估计提供具有一致性保证的插件实现。所得区间宽度可达预期的时间一致最优速率：非退化情形下为$\sqrt{\log\log n/n}$，退化情形下为$\log\log n/n$。该框架下讨论了几种广泛使用的U统计量，数值实验进一步验证了导出理论的有效性。