Open Access
Translator Disclaimer
April, 1993 Weak Convergence and Glivenko-Cantelli Results for Weighted Empirical $U$-Processes
Wilhelm Schneemeier
Ann. Probab. 21(2): 1170-1184 (April, 1993). DOI: 10.1214/aop/1176989287


Empirical processes of $U$-statistic structure were introduced by Serfling and studied in detail by Silverman, who proved weak convergence of weighted versions in the i.i.d. case. Our main theorem shows that this result can be generalized in two directions: First, the i.i.d. assumption can be omitted, and second, our proofs holds for a richer class of weight functions. In addition, we obtain almost sure convergence of weighted $U$-processes in the i.i.d. case which improves the results of Helmers, Janssen and Serfling, Aerts, Janssen and Mason and (in the special situation of the real line) Nolan and Pollard.


Download Citation

Wilhelm Schneemeier. "Weak Convergence and Glivenko-Cantelli Results for Weighted Empirical $U$-Processes." Ann. Probab. 21 (2) 1170 - 1184, April, 1993.


Published: April, 1993
First available in Project Euclid: 19 April 2007

zbMATH: 0776.60046
MathSciNet: MR1217585
Digital Object Identifier: 10.1214/aop/1176989287

Primary: 60F17

Keywords: $\mathscr{L}_b$-convergence , $U$-process , empirical process , pseudometric , ‎weight function

Rights: Copyright © 1993 Institute of Mathematical Statistics


Vol.21 • No. 2 • April, 1993
Back to Top