Could you help me, please, with the following problem ?

Let $(x_{i,j})_{(i,j)\in\mathbb{N}^2}$ be random variables verifying the two following properties :

* for any $(i,j)\in\mathbb{N}, x_{i,j} = \overline{x_{j,i}}$ 

* for any $(i,j) \neq (l,k)$ or $(k,l)$ the variables $x_{i,j}$ and $x_{l,k}$ are independant.

Do you know if there exists any formula which could help me to compute the following variance, by interchanging $\mathbb{V}$ and $\Sigma$ ?

$$\mathbb{V}\left[ \sum_{i_1,\dots,i_k=1}^n x_{i_1,i_2} x_{i_2,i_3} \dots x_{i_{k-1},i_k} x_{i_k,i_1} \right]$$

where $k$ is even.

It corresponds to the Variance of the Trace, $\mathbb{V}\left[\text{Tr}\left(X\right)\right]$ of a Hermitian random matrix $X=[x_{i,j}]_{(i,j)\in\{1,\dots,n\}^2}$.

More precisely, I am searching for an extension of this kind of formulas, which permit to interchange $\mathbb{V}$ and $\Sigma$ thanks to a decomposition of the initial sum into many independent sums :


$$\mathbb{V}\left[ \sum_{i,j=1}^n x_{i,j} \ x_{j,i} \right] = \mathbb{V}\left[ \sum_{i=1}^n |x_{ii}|^2 + 2\sum_{1\leq i<j\leq n} x_{i,j} \ x_{j,i} \right] =  \sum_{i=1}^n \mathbb{V}\left[|x_{ii}|^2\right] + 2\sum_{1\leq i<j\leq n} \mathbb{V}\left[ x_{i,j} \ x_{j,i} \right] $$

Thank you for your help.