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VOL. 5 | 2009 Interpolation spaces and the CLT in Banach spaces
Jim Kuelbs, Joel Zinn

Editor(s) Christian Houdré, Vladimir Koltchinskii, David M. Mason, Magda Peligrad


Necessary and sufficient conditions for the classical central limit theorem (CLT) for i.i.d. random vectors in an arbitrary separable Banach space require not only assumptions on the original distribution, but also on the sample. What we do here is to continue our study of the CLT in terms of the original distribution. Of course, some new ingredient must be introduced, so we allow slight modifications of the random vectors. In particular, we restrict our modifications to be continuous, and to be no larger than a fixed small number, or in some cases a fixed small proportion of the magnitude of the individual elements of the sample. We find that if we use certain interpolation space norms to measure the magnitude of such modifications, then the CLT can be improved. Examples of our result are also included.


Published: 1 January 2009
First available in Project Euclid: 2 February 2010

zbMATH: 1243.60023
MathSciNet: MR2797941

Digital Object Identifier: 10.1214/09-IMSCOLL506

Primary: 60F05
Secondary: 60F17

Keywords: best approximations , central limit theorems , interpolation spaces

Rights: Copyright © 2009, Institute of Mathematical Statistics


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