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

AbstractIn this paper we consider the weak convergence of the weighted multivariate empiricalU-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. Inference32, 1992, 259–269) from the i.i.d. case to the dependent case. M. Harel, C. A. O'Cinneide and M. L. Puri (J. Multivariate Anal.48, 1994, 297–314) gave only a generalization to the univariate case of the weighted empiricalU-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