A Cram\'er-Wold theorem for mixtures

Ricardo Fraiman; Leonardo Moreno; Thomas Ransford

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arXiv · 2410.22038

A Cram\'er-Wold theorem for mixtures

Ricardo Fraiman·Leonardo Moreno·Thomas Ransford

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Abstract

We show how a Cram\'er-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 Cram\'er-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.

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Ricardo Fraiman, Leonardo Moreno, Thomas Ransford. 2024-10-29. A Cram\'er-Wold theorem for mixtures. https://arxiv.org/abs/2410.22038

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