The Annals of Probability
- Ann. Probab.
- Volume 21, Number 2 (1993), 1170-1184.
Weak Convergence and Glivenko-Cantelli Results for Weighted Empirical $U$-Processes
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.
Ann. Probab., Volume 21, Number 2 (1993), 1170-1184.
First available in Project Euclid: 19 April 2007
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 60F17: Functional limit theorems; invariance principles
Schneemeier, Wilhelm. Weak Convergence and Glivenko-Cantelli Results for Weighted Empirical $U$-Processes. Ann. Probab. 21 (1993), no. 2, 1170--1184. doi:10.1214/aop/1176989287. https://projecteuclid.org/euclid.aop/1176989287