## The Annals of Probability

### A Remark on Convergence in Distribution of $U$-Statistics

#### Abstract

It is proved that, for $h$ measurable and symmetric in its arguments and $X_i$ i.i.d., if the sequence $\{n^{-m/2} \sum_{i_1,\ldots,i_m\leq n,i_j\neq i_k \text{if} j\neq k} h(X_{i_1},\ldots, X_{i_m})\}^\infty_{n=1}$ is stochastically bounded, then $Eh^2 < \infty$ and $Eh(X_1,x_2,\ldots,x_m) = 0$ a.s.

#### Article information

Source
Ann. Probab., Volume 22, Number 1 (1994), 117-125.

Dates
First available in Project Euclid: 19 April 2007

https://projecteuclid.org/euclid.aop/1176988850

Digital Object Identifier
doi:10.1214/aop/1176988850

Mathematical Reviews number (MathSciNet)
MR1258868

Zentralblatt MATH identifier
0801.60015

JSTOR
Gine, Evarist; Zinn, Joel. A Remark on Convergence in Distribution of $U$-Statistics. Ann. Probab. 22 (1994), no. 1, 117--125. doi:10.1214/aop/1176988850. https://projecteuclid.org/euclid.aop/1176988850