Uniform bound in the central limit theorem for Banach space valued dependent random variables

Let F be a Banach space with a sufficiently smooth norm. Let (Xi)i<=n be a sequence in LF2, and T be a Gaussian random variable T which has the same covariance as X = [Sigma]i<=nXi. Assume that there exists a constant G such that for s, [delta]>=0, we have P(s[less-than-or-equals, slant]||T||[less-than-or-equals, slant]s+[delta])[less-than-or-equals, slant]G[delta]. (*) We then give explicit bounds of [Delta](X) = supiP(X<=t)-P(T<=t) in terms of truncated moments of the variables Xi. These bounds hold under rather mild weak dependence conditions of the variables. We also construct a Gaussian random variable that violates (*).