The Annals of Probability

The Law of Large Numbers and the Central Limit Theorem in Banach Spaces

J. Hoffmann-Jorgensen and G. Pisier

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Abstract

Let $X_1, X_2, \cdots$ be independent random variables with values in a Banach space $E$. It is then shown that Chung's version of the strong law of large numbers holds, if and only if $E$ is of type $p$. If the $X_n$'s are identically distributed, then it is shown that the central limit theorem is valid, if and only if $E$ is of type 2. Similar results are obtained for vectorvalued martingales.

Article information

Source
Ann. Probab. Volume 4, Number 4 (1976), 587-599.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
http://projecteuclid.org/euclid.aop/1176996029

Digital Object Identifier
doi:10.1214/aop/1176996029

Mathematical Reviews number (MathSciNet)
MR423451

Zentralblatt MATH identifier
0368.60022

JSTOR
links.jstor.org

Subjects
Primary: 60F05: Central limit and other weak theorems
Secondary: 60B10: Convergence of probability measures 46E15: Banach spaces of continuous, differentiable or analytic functions

Keywords
Central limit theorem law of large numbers Banach space valued random variables martingales Banach space type modulus of uniform smoothness

Citation

Hoffmann-Jorgensen, J.; Pisier, G. The Law of Large Numbers and the Central Limit Theorem in Banach Spaces. Ann. Probab. 4 (1976), no. 4, 587--599. doi:10.1214/aop/1176996029. http://projecteuclid.org/euclid.aop/1176996029.


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