Testing the independence of two random vectors where only one dimension is large

For testing the independence of two vectors with respective dimensions (Formula presented.) and (Formula presented.), the existing literature in high-dimensional statistics all assume that both dimensions (Formula presented.) and (Formula presented.) grow to infinity with the sample size. However, as evidenced in RNA-sequencing data analysis, it happens frequently that one of the dimension is quite small and the other quite large compared to the sample size. In this paper, we address this new asymptotic framework for the independence test. A new test procedure is introduced and its asymptotic normality is established when the vectors are normally distributed. A Monte-Carlo study demonstrates the consistency of the procedure and exhibits its superiority over some existing high-dimensional procedures. It is also shown that the procedure is robust against the normality assumption on the population vectors. Applied to a set of RNA-sequencing data, we obtain very convincing results on pairwise independence/dependence of gene isoform expressions as attested by prior knowledge established in that field.