On the weak convergence of the empirical conditional copula under a simplifying assumption (Q1749990)

The authors study the problem of estimating the copula of two random variables conditional on a third random variable \(X\). They make the additional assumption that the conditional copula does not depend on the variable \(X\) that is conditioned on (while the marginal distributions conditional on \(X\) may change depending on the value of \(X\)). Under mild assumptions, it is shown that the empirical conditional copula behaves like an oracle estimator that presupposes the knowledge of the marginal distribution. For a sample of independent random vectors, a weak invariance principle is shown: The centered and rescaled empirical conditional copula converges to a Gaussian process. The rate of convergence is \(n^{-1/2}\), as it would be for the empirical unconditional copula. The authors also illustrate their findings with some simulations.