Abstract: | 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 = Σi≤nXi. Assume that there exists a constant G such that for s, δ≥0, we have P(sTs+δ)Gδ. (*) We then give explicit bounds of Δ(X) = supi|P(|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 (*). |