## The Annals of Probability

### Asymptotic Normality and Subsequential Limits of Trimmed Sums

#### 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

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