Summary
In Banach spaces the rate of convergence in the Central Limit Theorem is of orderO(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.
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Research supported by SFB 343 at Bielefeld and by the Alexander von Humboldt Foundation and completed at the University of Bielefeld, FRG
Research supported by the SFB 343 at Bielefeld
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Bentkus, V., Götze, F. On smoothness conditions and convergence rates in the CLT in Banach spaces. Probab. Th. Rel. Fields 96, 137–151 (1993). https://doi.org/10.1007/BF01192130
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DOI: https://doi.org/10.1007/BF01192130