If $x_n$ is a discrete time random signal and is white Gaussian noise (ergodic and WSS) so
$$E[x_n x_{n+l}]=\sigma ^2 \delta (l)$$
and
$$E[x_n]=0$$
Where $n \in \mathbb{R}$ and $l\in\mathbb{R}$
then what is:
$$\sum_{i=0}^{N-1}\sum_{j=0}^{N-1} E[x_i^2 x_j^2]$$
The problem I'm having is that I don't know if $E[x_i^2 x_j^2]=E[x_i^2]E[j_i^2]\;\forall \;i \ne j$
The expectation values of the summands factorize for $i\ne j$ because the joint probability density $p(x_i, x_j)$ factorizes into a product of Gaussians for $x_i$ and $x_j$. So you are left with the expectation value of the fourth powers. -- Count Iblis
I do know how to prove $E[x_n^4]=3\sigma ^4$. For reference the answer is $$3\sigma^4 N +\sigma^4N(N-1)=\sigma^4N(N+2)$$ -- texasflood
A reference for $E[x_n^4]=3\sigma ^4$: Calculation of the n-th central moment of the normal distribution $\mathcal{N}(\mu,\sigma^2)$