The Annals of Probability
- Ann. Probab.
- Volume 4, Number 4 (1976), 587-599.
The Law of Large Numbers and the Central Limit Theorem in Banach Spaces
J. Hoffmann-Jorgensen and G. Pisier
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
https://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. https://projecteuclid.org/euclid.aop/1176996029
