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 Law of Large Numbers and the Central Limit Theorem in Banach Spaces." Ann. Probab. 4 (4) 587 - 599, August, 1976. https://doi.org/10.1214/aop/1176996029