Sharp Support Thresholds for Smeariness of Absolutely Continuous Measures on Spheres
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866916074453204992 |
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| author | Pal, Susovan |
| author_facet | Pal, Susovan |
| contents | 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 \(R_m<S_m\). 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 \(\varepsilon>0\) with \(S_m+\varepsilon<π\), 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2606_02144 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Sharp Support Thresholds for Smeariness of Absolutely Continuous Measures on Spheres Pal, Susovan Statistics Theory Primary: 62H11, Secondary: 60F05, 60D05, 53C22 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 \(R_m<S_m\). 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 \(\varepsilon>0\) with \(S_m+\varepsilon<π\), 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. |
| title | Sharp Support Thresholds for Smeariness of Absolutely Continuous Measures on Spheres |
| topic | Statistics Theory Primary: 62H11, Secondary: 60F05, 60D05, 53C22 |
| url | https://arxiv.org/abs/2606.02144 |