Measure Dependent Asymptotic Rate of the Mean: Geometrical Smeariness
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The central limit theorem (CLT) for the mean in Euclidean space features a normal limiting distribution and an asymptotic rate of n^-1/2 for all probability measures it applies to. We revisit the generalized CLT for the Fréchet mean on hyperspheres. It has been found by Eltzner and Huckemann (2019) that for some probability measures, the sample mean fluctuates around the population mean asymptotically at a scale n^-α with exponent α = 1/6 with a non-normal distribution. This is at first glance in analogy to the situation on a circle, described by Hotz and Huckemann (2015). In this article we show that the phenomenon on hyperspheres of higher dimension is qualitatively different, as it does not rely on topological, but geometrical properties on the space, namely on the curvature, not on probability mass near the cut locus. This also leads to the conjecture that probability measures for which the asymptotic rate of the mean is α = 1/6 are possible more generally in positively curved spaces.
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