A Cramér-Wold theorem for mixtures
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866908393859448832 |
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| author | Fraiman, Ricardo Moreno, Leonardo Ransford, Thomas |
| author_facet | Fraiman, Ricardo Moreno, Leonardo Ransford, Thomas |
| contents | We show how a Cramér-Wold theorem for a family of multivariate probability distributions can be used to generate a similar theorem for mixtures (convex combinations) of distributions drawn from the same family.
Using this abstract result, we establish a Cramér-Wold theorem for mixtures of multivariate Gaussian distributions. According to this theorem, two such mixtures can be distinguished by projecting them onto a certain predetermined finite set of lines, the number of lines depending only on the total number Gaussian distributions involved and on the ambient dimension. A similar result is also obtained for mixtures of multivariate $t$-distributions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_22038 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A Cramér-Wold theorem for mixtures Fraiman, Ricardo Moreno, Leonardo Ransford, Thomas Probability 60B11 We show how a Cramér-Wold theorem for a family of multivariate probability distributions can be used to generate a similar theorem for mixtures (convex combinations) of distributions drawn from the same family. Using this abstract result, we establish a Cramér-Wold theorem for mixtures of multivariate Gaussian distributions. According to this theorem, two such mixtures can be distinguished by projecting them onto a certain predetermined finite set of lines, the number of lines depending only on the total number Gaussian distributions involved and on the ambient dimension. A similar result is also obtained for mixtures of multivariate $t$-distributions. |
| title | A Cramér-Wold theorem for mixtures |
| topic | Probability 60B11 |
| url | https://arxiv.org/abs/2410.22038 |