## The Annals of Probability

### Weak Convergence and Glivenko-Cantelli Results for Weighted Empirical $U$-Processes

Wilhelm Schneemeier

#### Abstract

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.

#### Article information

Source
Ann. Probab., Volume 21, Number 2 (1993), 1170-1184.

Dates
First available in Project Euclid: 19 April 2007

https://projecteuclid.org/euclid.aop/1176989287

Digital Object Identifier
doi:10.1214/aop/1176989287

Mathematical Reviews number (MathSciNet)
MR1217585

Zentralblatt MATH identifier
0776.60046

JSTOR
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