## The Annals of Probability

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

#### 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

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