A Cram\'er-Wold theorem for mixtures

TL;DR

This paper extends the Cramér-Wold theorem to mixtures of multivariate distributions, enabling the distinction of such mixtures through finite projections, with specific results for Gaussian and t-distributions.

Contribution

It introduces a novel theorem for mixtures of distributions, generalizing the Cramér-Wold theorem to identify mixtures via finite projections.

Findings

Mixtures of Gaussian distributions can be distinguished by projections onto a finite set of lines.

A similar theorem applies to mixtures of multivariate t-distributions.

The number of lines depends only on the number of components and the dimension.

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.