Checking that $𝐸[𝑥𝑥′⊗𝑥𝑥′] \prec \Sigma\otimes I+I\otimes \Sigma$

I have a random variable with mean 0 and covariance $$\Sigma$$, and I need to check that the following condition is satisfied

$$2\Sigma\otimes \Sigma+\text{vec} \Sigma(\text{vec}\Sigma)'\prec \Sigma\otimes I+I\otimes \Sigma$$

Here $$A\prec B$$ means that $$B-A$$ is positive definite and $$\otimes$$ refers to Kronecker product.

1. Has anyone seen this kind of relation come up in statistics? The left hand side is equal to $$E[xx'\otimes xx']$$ for a Gaussian random variable $$x$$

2. It can be solved with with generic eigenvalue solver by expanding out Kronecker products, but this makes the matrices quite big, is there an approach that takes into account the structure of the problem?

• in your scenario can you assume x is Gaussian, or you were commenting for another reason? – Lucas Roberts Sep 12 '20 at 21:54
• yes, I'm assuming that x comes from Gaussian centered at 0 – Yaroslav Bulatov Sep 12 '20 at 22:34
• The matrices here are growing quadratically in the size of the problem. One thing you could do is pivot to a triangular matrix and check the diagonal entries for non-negativity if the dimension growing quadratically is the problem. It doesn't mean that this doesn't show up in statistics but I haven't seen this Loewner ordering criteria for Gaussian matrices before. – Lucas Roberts Sep 13 '20 at 14:58
• One idea I had was to treat this as generalized eigenvalue problem (A,B), and use some kind of power method to solve this (because of structured form, multiplying/dividing by this matrices is fast) – Yaroslav Bulatov Sep 13 '20 at 15:16