Support of Continuous Smeary Measures on Spheres
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866915880718303232 |
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| author | Pal, Susovan |
| author_facet | Pal, Susovan |
| contents | We investigate the support of smeary, directionally smeary, and finite sample smeary probability measures $μ$ with density $ρ$ on spheres $\mathbb{S}^m$.
First, in the rotationally symmetric case, we show that a distribution is not smeary, or equivalently, not directionally smeary whenever its support lies in a geodesic ball centered at the Fréchet mean of radius $R_m>π/2$, where $R_m=π/2+O(1/m)$. In the general case, we show that neither directional nor full smeariness holds whenever the support is contained in a closed ball of radius $π/2$, however, past the support radius $π/2,$ full smeariness may break down, but directional smeariness breaks down only past the support radius $R_m.$
Second, we prove sharpness of this threshold. For every $\varepsilon>0$, we show there exists $m_0(\varepsilon)$ such that for all $m\ge m_0(\varepsilon)$ there exists a rotationally symmetric continuous smeary probability measure on $\mathbb{S}^m$ whose support lies in a ball of radius $π/2+\varepsilon$ around the Fréchet mean.
Third, in every dimension we construct directionally smeary continuous distributions supported in a ball of radius $π/2+\varepsilon$ whose Fréchet function has Hessian of rank one.
Finally, we study finite sample smeariness. We show that any continuous non-smeary distribution supported in a geodesic ball of radius $π/2$ is necessarily Type~I finite sample smeary, i.e. its variance modulation $m_n$ satisfies $\lim_{n\to\infty} m_n>1$. In the rotationally symmetric case, we further prove a curse-of-dimensionality phenomenon: the variance modulation increases with the dimension and can become arbitrarily large depending on the support. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_20974 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Support of Continuous Smeary Measures on Spheres Pal, Susovan Statistics Theory Primary: 62H11, Secondary: 60F05, 60D05, 53C22 We investigate the support of smeary, directionally smeary, and finite sample smeary probability measures $μ$ with density $ρ$ on spheres $\mathbb{S}^m$. First, in the rotationally symmetric case, we show that a distribution is not smeary, or equivalently, not directionally smeary whenever its support lies in a geodesic ball centered at the Fréchet mean of radius $R_m>π/2$, where $R_m=π/2+O(1/m)$. In the general case, we show that neither directional nor full smeariness holds whenever the support is contained in a closed ball of radius $π/2$, however, past the support radius $π/2,$ full smeariness may break down, but directional smeariness breaks down only past the support radius $R_m.$ Second, we prove sharpness of this threshold. For every $\varepsilon>0$, we show there exists $m_0(\varepsilon)$ such that for all $m\ge m_0(\varepsilon)$ there exists a rotationally symmetric continuous smeary probability measure on $\mathbb{S}^m$ whose support lies in a ball of radius $π/2+\varepsilon$ around the Fréchet mean. Third, in every dimension we construct directionally smeary continuous distributions supported in a ball of radius $π/2+\varepsilon$ whose Fréchet function has Hessian of rank one. Finally, we study finite sample smeariness. We show that any continuous non-smeary distribution supported in a geodesic ball of radius $π/2$ is necessarily Type~I finite sample smeary, i.e. its variance modulation $m_n$ satisfies $\lim_{n\to\infty} m_n>1$. In the rotationally symmetric case, we further prove a curse-of-dimensionality phenomenon: the variance modulation increases with the dimension and can become arbitrarily large depending on the support. |
| title | Support of Continuous Smeary Measures on Spheres |
| topic | Statistics Theory Primary: 62H11, Secondary: 60F05, 60D05, 53C22 |
| url | https://arxiv.org/abs/2603.20974 |