## The Annals of Probability

- Ann. Probab.
- Volume 2, Number 1 (1974), 90-102.

### Weak Convergence of Generalized $U$-Statistics

#### Abstract

Wichura (1969) studied an invariance principle for partial sums of a multi-dimensional array of independent random variables. It is shown that a similar invariance principle holds for a broad class of generalized $U$-statistics for which the different terms in the partial sums are not independent. Weak convergence of generalized $U$-statistics for random sample sizes is also studied. The case of (generalized) von Mises' functional is treated briefly.

#### Article information

**Source**

Ann. Probab., Volume 2, Number 1 (1974), 90-102.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176996754

**Digital Object Identifier**

doi:10.1214/aop/1176996754

**Mathematical Reviews number (MathSciNet)**

MR402844

**Zentralblatt MATH identifier**

0276.60008

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60B10: Convergence of probability measures

Secondary: 60G99: None of the above, but in this section

**Keywords**

$D \{\lbrack 0, 1 \rbrack^c\}$ space Generalized $U$-statistics invariance principle Gaussian processes relative compactness random indices Von Mises' differentiable statistical functions and weak convergence

#### Citation

Sen, Pranab Kumar. Weak Convergence of Generalized $U$-Statistics. Ann. Probab. 2 (1974), no. 1, 90--102. doi:10.1214/aop/1176996754. https://projecteuclid.org/euclid.aop/1176996754