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The Central Limit Theorem

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Probability in Banach Spaces

Part of the book series: Ergebnisse der Mathematik und ihrer Grenzgebiete ((CLASSICS,volume 23))

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Abstract

The study of strong limit theorems for sums of independent random variables such as the strong law of large numbers or the law of the iterated logarithm in the preceding chapters showed that in Banach spaces these can only be reasonably understood when the corresponding weak property, that is tightness or convergence in probability, is satisfied. It was shown indeed that under some natural moment conditions, the strong statements actually reduce to the corresponding weak ones. On the line, or in finite dimensional spaces, the moment conditions usually automatically ensure the weak limiting property. As we pointed out, this is no longer the case in general Banach spaces.

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Author information

Authors and Affiliations

  1. Institut de Recherche Mathématique Avancée, Département de Mathématique, Université Louis Pasteur, F-67084, Strasbourg, France

    Michel Ledoux

  2. Equipe d’Analyse, Université de Paris VI, F-75252, Paris, France

    Michel Talagrand

  3. Department of Mathematics, The Ohio State University, 43210, Columbus, OH, USA

    Michel Talagrand

Authors
  1. Michel Ledoux
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  2. Michel Talagrand
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© 1991 Springer-Verlag Berlin Heidelberg

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Ledoux, M., Talagrand, M. (1991). The Central Limit Theorem. In: Probability in Banach Spaces. Ergebnisse der Mathematik und ihrer Grenzgebiete, vol 23. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-20212-4_12

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  • DOI: https://doi.org/10.1007/978-3-642-20212-4_12

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-20211-7

  • Online ISBN: 978-3-642-20212-4

  • eBook Packages: Springer Book Archive

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