{"ID":2923722,"CreatedAt":"2026-06-02T04:05:25.881865328Z","UpdatedAt":"2026-06-04T07:41:34.29888543Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2606.02144","arxiv_id":"2606.02144","title":"Sharp Support Thresholds for Smeariness of Absolutely Continuous Measures on Spheres","abstract":"We investigate support thresholds for fully smeary and directionally smeary absolutely continuous probability measures on the sphere \\(\\mathbb{S}^m\\). The motivation is inferential: smeariness is caused by degeneracy of the Hessian of the Fréchet function, and such degeneracy can invalidate the classical central limit theorem (CLT) for Fréchet means and the corresponding Wald-type \\(χ^2\\) inference. For rotationally symmetric densities, we show that full and directional smeariness are equivalent. The Hessian and fourth-order terms are governed by two explicit geometry-dependent radii \\(R_m\u003cS_m\\). In dimensions \\(m=2,3\\), rotationally symmetric smeariness cannot occur. For \\(m\\ge4\\), support contained in the geodesic ball of radius \\(S_m\\), centered at the Fréchet mean, rules out smeariness; conversely, for every \\(\\varepsilon\u003e0\\) with \\(S_m+\\varepsilon\u003cπ\\), we construct examples of rotationally symmetric \\(2\\)-smeary densities supported in the ball of radius \\(S_m+\\varepsilon\\). For general densities, closed hemispherical support rules out both full and directional smeariness. Support contained in the closed ball of radius \\(S_m\\) rules out full smeariness, while we construct explicit, directionally \\(2\\)-smeary examples supported in balls of radius \\(π/2+\\varepsilon\\). As a byproduct, the explicit Hessian formulas in this paper also provide a practical diagnostic for detecting proximity to the Hessian-degenerate, non-classical regime.","short_abstract":"We investigate support thresholds for fully smeary and directionally smeary absolutely continuous probability measures on the sphere \\(\\mathbb{S}^m\\). The motivation is inferential: smeariness is caused by degeneracy of the Hessian of the Fréchet function, and such degeneracy can invalidate the classical central limit...","url_abs":"https://arxiv.org/abs/2606.02144","url_pdf":"https://arxiv.org/pdf/2606.02144v1","authors":"[\"Susovan Pal\"]","published":"2026-06-01T12:10:26Z","proceeding":"math.ST","tasks":"[\"math.ST\"]","methods":"[]","has_code":false}