• asymptotic representation;
  • conditional Kendall's tau;
  • empirical copula process;
  • fixed design;
  • random design;
  • smoothing;
  • weak convergence

Abstarct.  This paper is concerned with studying the dependence structure between two random variables Y1 and Y2 conditionally upon a covariate X. The dependence structure is modelled via a copula function, which depends on the given value of the covariate in a general way. Gijbels et al. (Comput. Statist. Data Anal., 55, 2011, 1919) suggested two non-parametric estimators of the ‘conditional’ copula and investigated their numerical performances. In this paper we establish the asymptotic properties of the proposed estimators as well as conditional association measures derived from them. Practical recommendations for their use are then discussed.