Arbitrary Rates of Convergence for Projected and Extrinsic Means

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.