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

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 t...

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Bibliographic Details
Published in:	The Annals of probability Vol. 4; no. 4; pp. 587 - 599
Main Authors:	Hoffmann-Jørgensen, J., Pisier, G.
Format:	Journal Article
Language:	English
Published:	Institute of Mathematical Statistics 01-08-1976 The Institute of Mathematical Statistics
Subjects:	46E15 60B10 60F05 Banach space Banach space type Banach space valued random variables Central limit theorem Law of large numbers Linear transformations Martingales Mathematical moments modulus of uniform smoothness Perceptron convergence procedure Radon Random variables Series convergence
Online Access:	Get full text
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Summary:	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.
ISSN:	0091-1798 2168-894X
DOI:	10.1214/aop/1176996029