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：我们考虑在序列设置中对两个单变量随机变量进行独立性检验的问题。通过利用安全且任意时刻有效推断的最新进展，我们提出了一种具有时间均匀一型误差控制的检验方法，并推导了该检验在有限样本性能及预期停止时间上的显式界。我们将该方法的实证性能与现有序列及非序列独立性检验进行了比较。此外，由于所提检验在原假设下是无分布的，我们通过实证模拟了维尔不等式（即马尔可夫不等式在鞅超中的对应形式）所造成的间隙——该不等式常用于控制任意时刻有效推断中的一型误差，并据此构造了一种截断序列检验。