We consider the problem of independence testing for two univariate random variables in a sequential setting. By leveraging recent developments on safe, anytime-valid inference, we propose a test with time-uniform type-I error control and derive explicit bounds on the finite sample performance of the test and the expected stopping time. We demonstrate the empirical performance of the procedure in comparison to existing sequential and non-sequential independence tests. Furthermore, since the proposed test is distribution free under the null hypothesis, we empirically simulate the gap due to Ville's inequality, the supermartingale analogue of Markov's inequality, that is commonly applied to control type I error in anytime-valid inference, and apply this to construct a truncated sequential test.

翻译：本文研究在序贯设定下对两个单变量随机变量进行独立性检验的问题。利用安全任意时刻有效推断领域的最新进展，我们提出一种具有时间一致I型错误控制的检验方法，推导了检验有限样本性能与期望停止时间的显式界。通过将所提方法与现有序贯及非序贯独立性检验进行实证对比，验证了其性能表现。此外，由于所提检验在原假设下具有分布无关性，我们通过数值模拟评估了Ville不等式（任意时刻有效推断中常用于控制I型误差的鞅类比马尔可夫不等式）所导致的间隙，并据此构建了截断序贯检验。