Uniform bound in the central limit theorem for Banach space valued dependent random variables (Q1084741)

The authors give a rate of convergence in C.L.T. for Banach space valued random variables with differentiable norm under dependence assumptions related to martingale differences. With p-moments assumptions the rate of convergence in Levy's metrics is \(O(n^{-p/6(1+p)})\). The main tool of this paper is theorem 5 in which the authors give a construction of \((T^ 1,...,T^ n)\) Gaussian r.v. having the same conditional covariance structure as previously given ones. Then classical Trotter's method allows to use Taylor's developments. The proof is completed supposing that the limiting Gaussian variable has norm with bounded density; as shown by the authors, this is not always the case but they give an alternative proof avoiding this difficulty. This technique was already used in \textit{P. L. Butzer}, \textit{L. Hahn} and \textit{M. Th. Roeckerath}, ibid. 13, 287-301 (1983; Zbl 0515.60014) showing a rate of convergence \(n^{-1/8}\) in the martingale difference case.