CouldCan you help me, please, with the following problem compute a variance?
Let $(x_{i,j})_{(i,j)\in\mathbb{N}^2}$ be random variables verifying thewith these 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 independantindependent.
Do you know ifIs there exists anya formula which could help mehelps to compute the following variance, for even $k$, 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 Variancevariance of the Tracetrace, $\mathbb{V}\left[\text{Tr}\left(X\right)\right]$$\mathbb{V}\left[\text{Tr}\left(X^k\right)\right]$ of a Hermitian random matrix $X=[x_{i,j}]_{(i,j)\in\{1,\dots,n\}^2}$.
More precisely, I am searchinglooking for an extension of this kind of formulas, which permit to interchangethe formula for interchanging $\mathbb{V}$ and $\Sigma$ thanks to a decomposition ofby decomposing the initial sum into many independent sums :
$$\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] $$$$\mathbb{V}\left[ \sum_{i,j=1}^n x_{i,j} \ x_{j,i} \right] = \sum_{i=1}^n \mathbb{V}\left[|x_{ii}|^2\right] + 2\sum_{1\leq i<j\leq n} \mathbb{V}\left[ x_{i,j} \ x_{j,i} \right]$$
Thank you for your help.
