Key points are not available for this paper at this time.
Let X₁, X₂, 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 Xₙ'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.
Hoffmann-Jørgensen et al. (Sun,) studied this question.