## The Annals of Probability

- Ann. Probab.
- Volume 20, Number 4 (1992), 1779-1804.

### Necessary and Sufficient Conditions for Asymptotic Normality of $L$-Statistics

David M. Mason and Galen R. Shorack

#### Abstract

It is now classical that the sample mean $\bar{Y}$ is known to be asymptotically normal with $\sqrt n$ norming if and only if $0 < \operatorname{Var}\lbrack Y\rbrack < \infty$ and with arbitrary norming if and only if the df of $Y$ is in the domain of attraction of the normal df. Now let $T_n = n^{-1}\sum c_{ni}h(X_{n:i})$ for order statistics $X_{n:i}$ from a $\operatorname{df} F$ denote a general $L$-statistic subject to a bit of regularity; the key condition introduced into this problem in this paper is the regular variation of the score function $J$ defining the $c_{ni}$'s. We now define a rv $Y$ by $Y = K(\xi)$, where $\xi$ is uniform (0, 1) and where $dK = J dh(F^{-1})$. Then $T_n$ is shown to be asymptotically normal with $\sqrt n$ norming if and only if $0 < \operatorname{Var}\lbrack Y\rbrack < \infty$ and with arbitrary norming if and only if the df of $Y$ is in the domain of attraction of the normal df. As it completely parallels the classical theorem, this theorem gives the right conclusion for $L$-statistics. In order to establish the necessity above, we also obtain a nice necessary and sufficient condition for the stochastic compactness of $T_n$ and give a representation formula for all possible subsequential limit laws.

#### Article information

**Source**

Ann. Probab., Volume 20, Number 4 (1992), 1779-1804.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176989529

**Digital Object Identifier**

doi:10.1214/aop/1176989529

**Mathematical Reviews number (MathSciNet)**

MR1188042

**Zentralblatt MATH identifier**

0765.62024

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60F05: Central limit and other weak theorems

Secondary: 62G30: Order statistics; empirical distribution functions 62F05: Asymptotic properties of tests 62E20: Asymptotic distribution theory 62E10: Characterization and structure theory

**Keywords**

$L$-statistics regularly varying stochastic compactness

#### Citation

Mason, David M.; Shorack, Galen R. Necessary and Sufficient Conditions for Asymptotic Normality of $L$-Statistics. Ann. Probab. 20 (1992), no. 4, 1779--1804. doi:10.1214/aop/1176989529. https://projecteuclid.org/euclid.aop/1176989529