On smoothness conditions and convergence rates in the CLT in Banach spaces (Q1326353)

In Banach spaces the rate of convergence in the central limit theorem is of order \(O(n^{-1/2})\) for sets which have `regular' boundaries with respect to the given covariance structure and which are three times differentiable. We show that in infinite-dimensional spaces it is impossible to weaken this differentiability condition in general, whereas in finite-dimensional spaces the assumption of convexity suffices. Similar results hold for the expectation of smooth functionals.