Weak convergence of empirical copula processes

Abstract

Weak convergence of the empirical copula process has been established by Deheuvels in the case of independent marginal distributions. Van der Vaart and Wellner utilize the functional delta method to show convergence in ${\ell }^{\mathrm{\infty }}\left(\right[a,b{]}^2)$ for some 0<a<b<1, under restrictions on the distribution functions. We extend their results by proving the weak convergence of this process in ${\ell }^{\mathrm{\infty }}\left(\right[0,1{]}^2)$ under minimal conditions on the copula function, which coincides with the result obtained by Gaenssler and Stute. It is argued that the condition on the copula function is necessary. The proof uses the functional delta method and, as a consequence, the convergence of the bootstrap counterpart of the empirical copula process follows immediately. In addition, weak convergence of the smoothed empirical copula process is established.