Arbitrary Rates of Convergence for Projected and Extrinsic Means

10/24/2019

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by   Christof Schötz, et al.

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We study central limit theorems for the projected sample mean of independent and identically distributed observations on subsets Q ⊂R^2 of the Euclidean plane. It is well-known that two conditions suffice to obtain a parametric rate of convergence for the projected sample mean: Q 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 Q. We show that breaking one of these conditions can lead to any other rate: For a virtually arbitrary prescribed rate, we construct Q such that all distributions with expectation at a preassigned point attain this rate.