## The Annals of Mathematical Statistics

- Ann. Math. Statist.
- Volume 29, Number 1 (1958), 118-130.

### A Central Limit Theorem for Sums of Interchangeable Random Variables

#### Abstract

A collection of random variables is defined to be interchangeable if every finite subcollection has a joint distribution which is a symmetric function of its arguments. Double sequences of random variables $X_{nk}, k = 1, 2, \cdots, k_n (\rightarrow \infty), n = 1, 2, \cdots$, interchangeable (as opposed to independent) within rows, are considered. For each $n, X_{n1}, \cdots, X_{n,k_n}$ may (a) have a non-random sum, or (b) be embeddable in an infinite sequence of interchangeable random variables, or (c) neither. In case (a), a theorem is obtained providing conditions under which the partial sums have a limiting normal distribution. Applications to such well-known examples as ranks and percentiles are exhibited. Case (b) is treated elsewhere while case (c) remains open.

#### Article information

**Source**

Ann. Math. Statist., Volume 29, Number 1 (1958), 118-130.

**Dates**

First available in Project Euclid: 27 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aoms/1177706709

**Digital Object Identifier**

doi:10.1214/aoms/1177706709

**Mathematical Reviews number (MathSciNet)**

MR93815

**Zentralblatt MATH identifier**

0085.35102

**JSTOR**

links.jstor.org

#### Citation

Chernoff, H.; Teicher, H. A Central Limit Theorem for Sums of Interchangeable Random Variables. Ann. Math. Statist. 29 (1958), no. 1, 118--130. doi:10.1214/aoms/1177706709. https://projecteuclid.org/euclid.aoms/1177706709