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.
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Hoffmann-Jørgensen et al. (1976) studied this question.