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Weak convergence of weighted multivariate empirical U-statistics processes under mixing condition. (English) Zbl 0894.60024

Summary: Recently, the asymptotic behavior of weighted multivariate empirical U-statistic was studied by F. H. Ruymgaart and M. C. A. van Zuijlen [ibid. 32, No. 2, 259-269 (1992; Zbl 0754.62032)] when the random variables are i.i.d. and by M. Harel, C. A. O’Cinneide and M. L. Puri [J. Multivariate Anal. 48, No. 2, 297-314 (1994; Zbl 0793.60021)] when the random variables are univariate and dependent. Our paper extends their results to the case when the U-statistic is indexed by \(t\in \mathbb{R}^p\) and the variables are dependent. We have to mention that the asymptotic behavior of U-statistics was studied when the random variables are absolutely regular and stationary by K. Yoshihara [Z. Wahrscheinlichkeitstheorie Verw. Geb. 35, 237-252 (1976; Zbl 0314.60028)] and when the variables are not stationary by M. Harel and M. L. Puri [J. Multivariate Anal. 30, No. 2, 181-204 (1989; Zbl 0683.60007), ibid. 31, No. 2, 258-265 (1989; Zbl 0693.62023) and Stochastic Processes Appl. 34, No. 2, 341-360 (1990; Zbl 0701.60025)].

MSC:

60F05 Central limit and other weak theorems
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