Weak convergence from projected laws on a positive-measure set of directions
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866910124494290944 |
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| author | Cholaquidis, Alejandro Banadik, Manuel Hernandez |
| author_facet | Cholaquidis, Alejandro Banadik, Manuel Hernandez |
| contents | The Cramér-Wold device characterises weak convergence of probability measures on $\mathbb{R}^d$ through convergence of all one-dimensional projected laws. We prove that, if the target projected laws are moment-determinate for surface-almost every direction, then weak convergence already follows from projected convergence on a positive-measure set of directions. This yields a simple probabilistic interpretation: if one samples a direction at random from any distribution on the sphere that is absolutely continuous with respect to surface measure, then, with probability one, convergence of the projected law along the sampled direction already forces global weak convergence under the same moment-determinacy assumption. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_10310 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Weak convergence from projected laws on a positive-measure set of directions Cholaquidis, Alejandro Banadik, Manuel Hernandez Probability Statistics Theory Other Statistics The Cramér-Wold device characterises weak convergence of probability measures on $\mathbb{R}^d$ through convergence of all one-dimensional projected laws. We prove that, if the target projected laws are moment-determinate for surface-almost every direction, then weak convergence already follows from projected convergence on a positive-measure set of directions. This yields a simple probabilistic interpretation: if one samples a direction at random from any distribution on the sphere that is absolutely continuous with respect to surface measure, then, with probability one, convergence of the projected law along the sampled direction already forces global weak convergence under the same moment-determinacy assumption. |
| title | Weak convergence from projected laws on a positive-measure set of directions |
| topic | Probability Statistics Theory Other Statistics |
| url | https://arxiv.org/abs/2604.10310 |