In this paper the question of the extent to which truncated heavy tailed random vectors, taking values in a Banach space, retain the characteristic features of heavy tailed random vectors, is answered from the point of view of the central limit theorem.
"Central Limit Theorem for truncated heavy tailed Banach valued random vectors." Electron. Commun. Probab. 15 346 - 364, 2010. https://doi.org/10.1214/ECP.v15-1564