- Discussion paper:
- arXiv:1910.11223
- Title:
*Arbitrary rates of convergence for projected and extrinsic means*- Author:
- Christof Schötz
- Abstract:
- We study central limit theorems for the projected sample mean of independent and identically distributed observations on subsets M of the Euclidean plane R^2. It is well-known that two conditions suffice to obtain a parametric rate of convergence for the projected sample mean: M is a C^2-manifold, and the expectation of the underlying distribution calculated in R^2 is bounded away from the medial axis, the set of point that do not have a unique projection to M. We show that breaking one of these conditions can lead to any other rate: For a virtually arbitrary prescribed rate, we construct M such that all distributions with expectation at a preassigned point attain this rate.

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