## The Annals of Probability

- Ann. Probab.
- Volume 20, Number 1 (1992), 455-482.

### Asymptotic Behavior of Self-Normalized Trimmed Sums: Nonnormal Limits

Marjorie G. Hahn and Daniel C. Weiner

#### Abstract

Let $\{X_j\}$ be independent, identically distributed random variables with continuous nondegenerate distribution $F$ which is symmetric about the origin. Let $\{X_n(1), X_n(2),\ldots, X_n(n)\}$ denote the arrangement of $\{X_1,\ldots, X_n\}$ in decreasing order of magnitude, so that with probability 1, $|X_n(1)| > |X_n(2)| > \cdots > |X_n(n)|$. For integers $r_n \rightarrow \infty$ such that $r_n/n \rightarrow 0$, define the self-normalized trimmed sum $T_n = \sum^n_{i=r_n}X_n(i)/\{\sum^n_{i=r_n}X^2_n(i)\}^{1/2}$. The asymptotic behavior of $T_n$ is studied. Under a probabilistically meaningful analytic condition generalizing the asymptotic normality criterion for $T_n$, various interesting nonnormal limit laws for $T_n$ are obtained and represented by means of infinite random series. In general, moreover, criteria for degenerate limits and stochastic compactness for $\{T_n\}$ are also obtained. Finally, more general results and technical difficulties are discussed.

#### Article information

**Source**

Ann. Probab., Volume 20, Number 1 (1992), 455-482.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176989937

**Digital Object Identifier**

doi:10.1214/aop/1176989937

**Mathematical Reviews number (MathSciNet)**

MR1143431

**Zentralblatt MATH identifier**

0743.60024

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60F05: Central limit and other weak theorems

Secondary: 62G05: Estimation 62G30: Order statistics; empirical distribution functions

**Keywords**

Trimmed sums self-normalization and studentization magnitude order statistics stochastic compactness weak convergence series representations symmetry nonnormal limits infinitely divisible laws

#### Citation

Hahn, Marjorie G.; Weiner, Daniel C. Asymptotic Behavior of Self-Normalized Trimmed Sums: Nonnormal Limits. Ann. Probab. 20 (1992), no. 1, 455--482. doi:10.1214/aop/1176989937. https://projecteuclid.org/euclid.aop/1176989937