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March, 1958 A Central Limit Theorem for Sums of Interchangeable Random Variables
H. Chernoff, H. Teicher
Ann. Math. Statist. 29(1): 118-130 (March, 1958). DOI: 10.1214/aoms/1177706709


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.


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H. Chernoff. H. Teicher. "A Central Limit Theorem for Sums of Interchangeable Random Variables." Ann. Math. Statist. 29 (1) 118 - 130, March, 1958.


Published: March, 1958
First available in Project Euclid: 27 April 2007

zbMATH: 0085.35102
MathSciNet: MR93815
Digital Object Identifier: 10.1214/aoms/1177706709

Rights: Copyright © 1958 Institute of Mathematical Statistics

Vol.29 • No. 1 • March, 1958
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