We show how a recent theorem of J. Hoffmann-Jorgensen and G. Pisier can be formulated in such a way as to include a theorem of N. Jain and M. Marcus. We also obtain some central limit theorems on $L^p\lbrack 0, 1\rbrack$, for $1 \leqq p < 2$.
"A Note on the Central Limit Theorem in Banach Spaces." Ann. Probab. 5 (2) 283 - 286, April, 1977. https://doi.org/10.1214/aop/1176995852