I was trying to find the variance of a quadratic form $\underline{x}^{T} A \underline{x}$ where it is given that
$$\mathbb E(X_i)=\nu_i, \quad \mbox{Var}(X_i)=\mu_{2}, \quad \mathbb E(X_i-\nu_i)^3= \mu_3$$
and so on. It is also provided that $A$ is symmetric and non-stochastic.
I first centralized the variables by taking $Y_i=X_i-\nu_i$ And after tedious calculations I found the $Var(\underline{y}^{T}A\underline{y})=\sum_{i=1}^{n}a_{ii}^2 \mu_{4}+\mu_{2}^2[(\sum a_{ii})^2-3 \sum a_{ii}^2 +2 \sum \sum a_{ij}^2]-(\sum a_{ii})^2$. But as I need to find $Var(\underline{x}^{T}A\underline{x})$, I need to calculate some expectations of additional quantities like $E(\underline{x}^{T}A\underline{x})(\underline{x}^{T}A \underline{\nu})$ which I am finding difficult to compute.
Can you offer a way out how to calculate $E(\underline{x}^{T}A\underline{x})(\underline{x}^{T}A \underline{\nu})$ ?
Or , you can give a new approach instead. Thanks!