Bernoulli 11(5), 2005, 829-845. It has recently come to our attention that there is a large overlap between our Theorem 1.2 and the work of Tanaka (1973) and Cuesta and Matran (1989). In particular, the same result as ours is proved in both cases by analysing the same subadditive sequence, for real-valued random variables in Tanaka (1973) and in the more general Hilbert space setting in Cuesta and Matran (1989). Both these papers assume the stronger condition of a finite fourth moment, though they both comment that this can be weakened, perhaps to just requiring a finite (2 + δ)th moment, for δ > 0. In our paper we need only a finite second moment. We acknowledge the priority of the above authors on this result and sincerely apologize for our oversight.