For geodesically symmetric heavy-tailed distributions on non-compact symmetric spaces, the sample Fréchet mean converges in probability to the center of symmetry under the Kolmogorov-Feller tail condition.
General M-estimators of location on Riemannian manifolds: existence and uniqueness
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abstract
We study general M-estimators of location on Riemannian manifolds, extending classical notions such as the Frechet mean by replacing the squared loss with a broad class of loss functions. Under minimal regularity conditions on the loss function and the underlying probability distribution, we establish theoretical guarantees for the existence and uniqueness of such estimators. In particular, we provide sufficient conditions under which the population and sample M-estimators exist and are uniquely defined. Our results offer a general framework for robust location estimation in non-Euclidean geometric spaces and unify prior uniqueness results under a broad class of convex losses.
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On the Kolmogorov-Feller weak law of large numbers for the Fr\'echet mean on non-compact symmetric spaces
For geodesically symmetric heavy-tailed distributions on non-compact symmetric spaces, the sample Fréchet mean converges in probability to the center of symmetry under the Kolmogorov-Feller tail condition.