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## Determining if a given probability distribution in the plane is a mixture of others

Suppose that $f_0(\cdot)$ is a probability distribution in the plane with a known mean $\mu_{0}\in\mathbb{R}^2$ and a covariance matrix $\Sigma_0\in\mathbb{R}^{2\times2}$ that is bounded by a linear matrix inequality, $S_0 - \Sigma_0 \succeq 0$, for some given (symmetric, positive semi-definite) $S_0$. Suppose we are given $n$ additional points $\mu_i\in\mathbb{R}^2$ and $n$ additional (symmetric, positive semi-definite) matrices $S_i\in\mathbb{R}^{2\times2}$. Is it easy to determine if there exist $n$ distributions $f_i(\cdot)$ such that:

1) $f_0(\cdot)$ can be expressed as a convex combination of the distributions $f_i(\cdot)$,

2) $\mu_i$ is the mean of $f_i(\cdot)$ for all $i$, and

3) $S_i - \Sigma_i \succeq 0$ for all $i$, where $\Sigma_i$ is the covariance matrix of $f_i(\cdot)$?

Is the problem easier if $f_0(\cdot)$ is discrete?

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