Measure Dependent Asymptotic Rate of the Mean: Geometrical and Topological Smeariness
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We revisit the generalized central limit theorem (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 exponents α < 1/2 with a non-normal distribution. This is at first glance in analogy to the situation on a circle. 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. This also leads to the expectation that probability measures for which the asymptotic rate of the mean is slower than α = 1/2 are possible more generally in positively curved spaces.
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