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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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

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

First available in Project Euclid: 19 April 2007

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Zentralblatt MATH identifier


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

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


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.

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