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Limit theorems for weakly exchangeable arrays

Published online by Cambridge University Press:  24 October 2008

G. K. Eagleson  and
N. C. Weber
G. K. Eagleson
Affiliation:
Statistical Laboratory, University of Cambridge
N. C. Weber
Affiliation:
Statistical Laboratory, University of Cambridge
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Extract

An array of random variables, indexed by a multidimensional parameter set, is said to be dissociated if the random variables are independent whenever their indexing sets are disjoint. The idea of dissociated random variables, which arises rather naturally in data analysis, was first studied by McGinley and Sibson(7). They proved a Strong Law of Large Numbers for dissociated random variables when their fourth moments are uniformly bounded. Silver man (8) extended the analysis of dissociated random variables by proving a Central Limit Theorem when the variables also satisfy certain symmetry relations. It is the aim of this paper to show that a Strong Law of Large Numbers (under more natural moment conditions), a Central Limit Theorem and in variance principle are consequences of the symmetry relations imposed by Silverman rather than the independence structure. To prove these results, reversed martingale techniques are employed and thus it is shown, in passing, how the well known Central Limit Theorem for U-statistics can be derived from the corresponding theorem for reversed martingales (as was conjectured by Loynes(6)).

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 84 , Issue 1 , July 1978 , pp. 123 - 130
Copyright
Copyright © Cambridge Philosophical Society 1978

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References

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