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The Law of Large Numbers and the Central Limit Theorem in Banach Spaces
J. Hoffmann-Jørgensen and G. Pisier
The Annals of Probability
Vol. 4, No. 4 (Aug., 1976), pp. 587-599
Published by: Institute of Mathematical Statistics
Stable URL: http://www.jstor.org/stable/2243043
Page Count: 13
You can always find the topics here!Topics: Law of large numbers, Radon, Random variables, Banach space, Central limit theorem, Martingales, Mathematical moments, Topological theorems, Linear transformations, Perceptron convergence procedure
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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.
The Annals of Probability © 1976 Institute of Mathematical Statistics