A rank-based sequential test of independence.

The article focuses on developing a new rank-based sequential test for independence between two univariate random variables, leveraging safe, anytime-valid inference to control Type-I error uniformly over time. The proposed method transforms observations into randomized sequential ranks, reducing the composite null hypothesis of independence to testing uniformity on the unit square, and constructs test martingales based on discretized bin frequencies with a multinomial model. The authors derive explicit finite-sample bounds on the test's power, demonstrate how to aggregate tests over multiple discretization levels, and provide practical corrections to improve finite-sample performance. Empirical comparisons show that the rank-based test with Sinkhorn correction generally outperforms existing sequential and nonsequential methods, and an application to weather-related data illustrates its utility in real-world sequential monitoring scenarios.