A variance computation

Question

Can you help me compute a variance?

Let $(x_{i,j})_{(i,j)\in\mathbb{N}^2}$ be random variables with these two 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 independent.

Is there a formula which helps to compute the following variance, for even $k$, by interchanging $\mathbb{V}$ and $\Sigma$?

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

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

I am looking for an extension of the formula for interchanging $\mathbb{V}$ and $\Sigma$ by decomposing the initial sum into many independent sums:

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

Thank you for your help.

Answer 1

For small $n$ you could just evaluate this by explicit calculation. For large $n$ there is a useful asymptotics:

Wigner [1] showed that, under quite general conditions on the distribution of the independent matrix elements, and setting their variance equal to unity, the large-$n$ asymptotics of the expectation value of $\text{Tr}\,(X^k)$ is given by a Catalan number, $$E\,[\text{Tr}\,(X^k)]=\frac{n}{s+1}{{2s}\choose{s}}+O(n^{-1}),\;\;k=2s\;\;\text{even}.$$ The distribution of this trace satisfies a central-limit-theorem, converging for large $n$ to a Gaussian distribution with variance $1/\pi$. See for example Central limit theorem for traces of large random symmetric matrices with independent matrix elements.

[1] E.P. Wigner, On the Distribution of the Roots of Certain Symmetric Matrices (1958).