Arbitrary Rates of Convergence for Projected and Extrinsic Means

Abstract

We study central limit theorems for the projected sample mean of independent and identically distributed observations on subsets $\mathcal{Q}\subset {\mathbb{R}}^2$ of the Euclidean plane. It is well-known that two conditions suffice to obtain a parametric rate of convergence for the projected sample mean: $\mathcal{Q}$ is a ${\mathcal{C}}^2$-manifold, and the expectation of the underlying distribution calculated in ${\mathbb{R}}^2$ is bounded away from the medial axis, the set of point that do not have a unique projection to $\mathcal{Q}$. We show that breaking one of these conditions can lead to any other rate: For a virtually arbitrary prescribed rate, we construct $\mathcal{Q}$ such that all distributions with expectation at a preassigned point attain this rate.

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