On the rate of convergence in the central limit theorem and the type of the Banach space

Let E be a Banach space. Let [xi] be a sequence of which goes to zero. Let X be a centered E-valued random variable, which is bounded. Let Sn be the sum of n independent copies of X. Assume that whenever X satisfies the CLT, we have. where [mu] is the (Gaussian) limit of the laws of Sn. Then E is type 2.