## The Annals of Mathematical Statistics

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

#### 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
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