I am looking at the 4th central moment of a weighted-sum of correlated random variables, which takes the form

$$\mu_4 = \sum_{i,j,k,l=1}^n w_i w_j w_k w_l \mu_{ijkl}$$

where $\mu_{ijkl}$ are the fourth-order co-moments of the $n$ random variables and $w_i$ are the weights. The variables I assume to be identically distributed and by correlated I mean that the dependence structure is defined by a Gaussian copula, so is a function of a correlation matrix only.

$\mu_4$ is a multivariate non-negative convex polynomial in the $w_i$ and is homogeneous of order 4. Numerically, individual cases can be written as a sum of squares, which can be obtained by solving a semi-definite program. As far as I am aware there does not exist any specific example of a non-negative convex polynomial that **cannot** be written as a sum-of-squares, so I believe this is always possible for $\mu_4$. I wondered if anyone might have an idea about how to prove this in general?

Thank you.