## The Annals of Probability

- Ann. Probab.
- Volume 17, Number 3 (1989), 1186-1219.

### Asymptotic Normality and Subsequential Limits of Trimmed Sums

Philip S. Griffin and William E. Pruitt

#### Abstract

Let $\{X_i\}$ be i.i.d. and $S_n(s_n, r_n)$ the sum of the first $n X_i$ with the $r_n$ largest and $s_n$ smallest excluded. Assume $r_n \rightarrow \infty, s_n \rightarrow \infty, n^{-1}r_n \rightarrow 0, n^{-1}s_n \rightarrow 0.$ Necessary and sufficient conditions are obtained for the existence of $\{\delta_n\}, \{\gamma_n\}$ such that $\gamma^{-1}_n(S_n(s_n, r_n) - \delta_n)$ converges weakly to a standard normal. The set of all subsequential limit laws for these sequences is characterized and sufficient conditions are given for $X_i$ to be in the domain of partial attraction of a given law in the class. These conditions are also necessary if a unique factorization result for characteristic functions is true.

#### Article information

**Source**

Ann. Probab., Volume 17, Number 3 (1989), 1186-1219.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176991264

**Digital Object Identifier**

doi:10.1214/aop/1176991264

**Mathematical Reviews number (MathSciNet)**

MR1009452

**Zentralblatt MATH identifier**

0688.60016

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60F05: Central limit and other weak theorems

**Keywords**

Asymptotic normality stochastic compactness discarding outliers

#### Citation

Griffin, Philip S.; Pruitt, William E. Asymptotic Normality and Subsequential Limits of Trimmed Sums. Ann. Probab. 17 (1989), no. 3, 1186--1219. doi:10.1214/aop/1176991264. https://projecteuclid.org/euclid.aop/1176991264