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Interpolation spaces and the CLT in Banach spaces

Jim Kuelbs and Joel Zinn

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Abstract

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.

Chapter information

Source
Christian Houdré, Vladimir Koltchinskii, David M. Mason and Magda Peligrad, eds., High Dimensional Probability V: The Luminy Volume (Beachwood, Ohio, USA: Institute of Mathematical Statistics, 2009), 73-83

Dates
First available in Project Euclid: 2 February 2010

Permanent link to this document
https://projecteuclid.org/euclid.imsc/1265119262

Digital Object Identifier
doi:10.1214/09-IMSCOLL506

Mathematical Reviews number (MathSciNet)
MR2797941

Zentralblatt MATH identifier
1243.60023

Subjects
Primary: 60F05: Central limit and other weak theorems
Secondary: 60F17: Functional limit theorems; invariance principles

Keywords
central limit theorems best approximations interpolation spaces

Rights
Copyright © 2009, Institute of Mathematical Statistics

Citation

Kuelbs, Jim; Zinn, Joel. Interpolation spaces and the CLT in Banach spaces. High Dimensional Probability V: The Luminy Volume, 73--83, Institute of Mathematical Statistics, Beachwood, Ohio, USA, 2009. doi:10.1214/09-IMSCOLL506. https://projecteuclid.org/euclid.imsc/1265119262


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References

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