Weak convergence for rectangle-indexed weighted multivariate empirical \(U\)-statistic processes under mixing conditions (Q1269080)

The author discusses the weak convergence of the weighted multivariate empirical \(U\)-statistics indexed by rectangles under mixing conditions in the Skorokhod topology. It is a generalization of a result of \textit{F. H. Ruymgaart} and subsequent extensions by \textit{M. C. A. van Zuijlen} [J. Stat. Plann. Inference 32, No. 2, 259-269 (1992; Zbl 0754.62032)], \textit{M. Harel}, \textit{C. A. O'Cinneide} and \textit{M. L. Puri} [J. Multivariate Anal. 48, No. 2, 297-314 (1994; Zbl 0793.60021)] and the author [J. Stat. Plann. Inference 61, No. 1, 1-16 (1997; Zbl 0894.60024)] in different directions. The author also points out many interesting applications of his result to Cramér-von Mises statistic or the Anderson-Darling statistic and to a Markov process as well as to an ARMA process.