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. 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：我们考虑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。通过利用安全、随时有效的推断领域的最新进展，我们提出了一种具有时间一致I类误差控制的检验方法，并推导了该检验有限样本性能的显式界。我们通过实验展示了该方法与现有序贯及非序贯独立性检验相比的经验性能。此外，由于所提出的检验在原假设下是无分布的，我们通过经验模拟了由Ville不等式（即马尔可夫不等式在超鞅中的类似形式，常用于控制随时有效推断中的I类误差）导致的差距，并据此构建了一种截断序贯检验。