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An almost everywhere central limit theorem

Published online by Cambridge University Press:  24 October 2008

Gunnar A. Brosamler
Gunnar A. Brosamler
Affiliation:
Fachbereich Mathematik, Universität des Saarlandes, Saarbrücken, West Germany Department of Mathematics, The University of British Columbia, Vancouver, B.C., Canada
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Extract

The purpose of this paper is the proof of an almost everywhere version of the classical central limit theorem (CLT). As is well known, the latter states that for IID random variables Y1, Y2, … on a probability space (Ω, , P) with we have weak convergence of the distributions of to the standard normal distribution on ℝ. We recall that weak convergence of finite measures μn on a metric space S to a finite measure μ on S is defined to mean that

for all bounded, continuous real functions on S. Equivalently, one may require the validity of (1·1) only for bounded, uniformly continuous real functions, or even for all bounded measurable real functions which are μ-a.e. continuous.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 104 , Issue 3 , November 1988 , pp. 561 - 574
Copyright
Copyright © Cambridge Philosophical Society 1988

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References

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