An asymptotic decomposition for multivariate distribution-free tests of independence

In the multivariate case, the empirical dependence function, defined as the empirical distribution function with reduced uniform margins on the unit interval, can be shown for an i.i.d. sequence to converge weakly in an asymptotic way to a limiting Gaussian process. The main result of this paper is that this limiting process can be canonically separated into a finite set of independent Gaussian processes, enabling one to test the existence of dependence relationships within each subset of coordinates independently (in an asymptotic way) of what occurs in the other subsets. As an application we derive the Karhunen-Loeve expansions of the corresponding processes and give the limiting distribution of the multivariate Cramer-Von Mises test of independence, generalizing results of Blum, Kiefer, Rosenblatt, and Dugué. Other extensions are mentioned, including a generalization of Kendall's τ.