## The Annals of Mathematical Statistics

- Ann. Math. Statist.
- Volume 43, Number 1 (1972), 31-41.

### Weak Convergence of $U$-Statistics and Von Mises' Differentiable Statistical Functions

R. G. Miller, Jr. and Pranab Kumar Sen

#### Abstract

For partial cumulative sums of independent and identically distributed random variables (i.i.d.r.v.) with a finite (positive) variance, weak convergence to Brownian motion processes has been established by Donsker (1951, 1952). The result is extended here to differentiable statistical functions of von Mises (1947) and $U$-statistics of Hoeffding (1948). Along with the extension to generalized $U$-statistics, a few applications are briefly sketched.

#### Article information

**Source**

Ann. Math. Statist., Volume 43, Number 1 (1972), 31-41.

**Dates**

First available in Project Euclid: 27 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aoms/1177692698

**Digital Object Identifier**

doi:10.1214/aoms/1177692698

**Mathematical Reviews number (MathSciNet)**

MR300321

**Zentralblatt MATH identifier**

0238.62057

**JSTOR**

links.jstor.org

#### Citation

Miller, R. G.; Sen, Pranab Kumar. Weak Convergence of $U$-Statistics and Von Mises' Differentiable Statistical Functions. Ann. Math. Statist. 43 (1972), no. 1, 31--41. doi:10.1214/aoms/1177692698. https://projecteuclid.org/euclid.aoms/1177692698