Exchangeability and irreducible rotational invariance
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866913883521810432 |
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| author | Baldi, Paolo Marinucci, Domenico Trapani, Stefano |
| author_facet | Baldi, Paolo Marinucci, Domenico Trapani, Stefano |
| contents | In this note we prove that a finite family $\{X_1,\dots,X_d\}$ of real r.v.'s that is exchangeable and such that $(X_1,\dots,X_d)$ is invariant with respect to a subgroup of $SO(d)$ acting irreducibly, is actually invariant with respect to the action of the full group $SO(d)$. Three immediate consequences are deduced: a characterization of isotropic spherical random eigenfunctions whose Fourier coefficients are exchangeable, an extension of Bernstein's characterization of the Gaussian and a characterization of the Lebesgue measure on the sphere. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2310_19611 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Exchangeability and irreducible rotational invariance Baldi, Paolo Marinucci, Domenico Trapani, Stefano Probability Representation Theory 20C30, 20G45, 22E46, 60B15, 60G60, 60G09 In this note we prove that a finite family $\{X_1,\dots,X_d\}$ of real r.v.'s that is exchangeable and such that $(X_1,\dots,X_d)$ is invariant with respect to a subgroup of $SO(d)$ acting irreducibly, is actually invariant with respect to the action of the full group $SO(d)$. Three immediate consequences are deduced: a characterization of isotropic spherical random eigenfunctions whose Fourier coefficients are exchangeable, an extension of Bernstein's characterization of the Gaussian and a characterization of the Lebesgue measure on the sphere. |
| title | Exchangeability and irreducible rotational invariance |
| topic | Probability Representation Theory 20C30, 20G45, 22E46, 60B15, 60G60, 60G09 |
| url | https://arxiv.org/abs/2310.19611 |