If $h_{1}, h_{2}, \dots, h_{N}$ are zero mean i.i.d Gaussian random variables with variance $\sigma_{h}^{2}$, then how do I calculate the value of
$$\mathbb{E}\left[\left(h_{1}^{2} + h_{2}^{2} + \dots, h_{N}^{2} \right)^{2}\right]$$
The only clue I have is that
$$\mathbb{E}\left[h_{i}^{2}\right] = \sigma_{h}^{2}$$
I do not know how to compute $\mathbb{E}\left[h_{i}^{2} h_{j}^{2}\right]$ (for all $i \neq j$). Please help.
On
Answer for the follow-up question in the comments
When you expand the square you will have a matrix of $(N-1)\times(N-1)$ elements.
The $N-1$ elements in the principal diagonal are in the form $E(h_i^2h_{i+1}^2)=\sigma^4$
All the other elements contains at least on element $E(h_i)=0$ thus they are all zero.
Thus the result is $(N-1)\sigma^4$
$$Y=h_1^2+\dots + h_N^2 =\sigma^2\left[\left(\frac{h_1}{\sigma}\right)^2+\dots+\left(\frac{h_N}{\sigma}\right)^2 \right]$$
It is self evident that
$$Y\sim Gamma\left(\frac{N}{2};\frac{1}{2\sigma^2}\right)$$
Thus its second moment is
$$\mathbb{E}[Y^2]=\mathbb{V}[Y]+\mathbb{E}^2[Y]=\frac{N}{2}4\sigma^4+\left(\frac{N}{2}2\sigma^2\right)^2=(N+2)N\sigma^4$$