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：我们研究了在序列设置中两个单变量随机变量的独立性检验问题。通过利用安全且任意时间有效推断的最新进展，我们提出了一种具有时间一致I类错误控制的检验方法，并推导了该检验有限样本性能及预期停止时间的显式界限。我们展示了该程序与现有序列及非序列独立性检验相比的实证性能。此外，由于所提检验在零假设下是分布无关的，我们通过实证模拟了Ville不等式（即Markov不等式的鞅类比）带来的间隙——该不等式常用于任意时间有效推断中的I类错误控制，并据此构建了一个截断序列检验。