Weak Convergence for Rectangle-Indexed Weighted Multivariate Empirical U-Statistic Processes under Mixing Conditions

In this paper we consider the weak convergence of the weighted multivariate empirical U-statistic indexed by rectangles under mixing conditions in the Skorohod topology. It is a generalization of a result of F. H. Ruymgaart and M. C. A. van Zuijlen J. Statist. Plann. Inference 32, 1992, 259]269.from the i.i.d. case to the dependent case. M. Harel, C. A. O’Cinneide and M. L. Puri J. Multi¨ariate Anal. 48, 1994, 297]314. gave only a generalization to the univariate case of the weighted empirical U-statistic indexed by point; a generalization to the multivariate case is obtained by B. Ragbi J. Statist. Plann. Inference, 61, 1997, 1]16.. There are a number of interesting applications dealing with the Cramer]Von Mises statistic or the Anderson]Darling statistic given in Section 4; applications to a Markov process as well as to an ARMA process are also provided.