First two moments of the elements of the scatter matrix Let $\mathbf{A}=\sum_{\alpha=1}^nx_\alpha x_\alpha^T$ and $x_\alpha$ is distributed according to $N(0,\Sigma)$. I want to show that the first two moments of the elements of $\mathbf{A}$ are
$$E\left[a_{ij}\right]=n\sigma_{ij}$$
$$E\left[\left(a_{ij}-n\sigma_{ij}\right)\left(a_{kl}-n\sigma_{kl}\right)\right]=n\left(\sigma_{il}\sigma_{jk}+\sigma_{ik}\sigma_{jl}\right)$$
I have no problem showing the first moment but I have some trouble showing the second moment.
Anderson (1963) is stating this without a proof or reference in his paper Asymptotic Theory for Principal Component Analysis
http://www.jstor.org/stable/2991288?seq=1#page_scan_tab_contents.
So I'm thinking that this is something quite straight forward or a standard result.
 A: Thank you, I managed to show this based on your comments
\begin{array}{l}
E\left[ {\left( {{a_{ij}} - n{\sigma _{ij}}} \right)\left( {{a_{kl}} - n{\sigma _{kl}}} \right)} \right] = E\left[ {{a_{ij}}{a_{kl}} - {a_{ij}}n{\sigma _{kl}} - n{\sigma _{ij}}{a_{kl}} + n{\sigma _{ij}}n{\sigma _{kl}}} \right] = \\
E\left[ {\sum\limits_{\alpha  = 1}^n {{x_{\alpha i}}x_{\alpha j}^{}} \sum\limits_{\alpha  = 1}^n {{x_{\alpha k}}x_{\alpha l}^{}} } \right] - {n^2}{\sigma _{ij}}{\sigma _{kl}} = \\
E\left[ {\sum\limits_{\alpha  = 1}^n {{x_{\alpha i}}x_{\alpha j}^{}{x_{\alpha k}}x_{\alpha l}^{}}  + \sum\limits_{\alpha  \ne \omega }^n {{x_{\alpha i}}x_{\alpha j}^{}{x_{\omega k}}x_{\omega l}^{}} } \right] - {n^2}{\sigma _{ij}}{\sigma _{kl}} = \\
E\left[ {\sum\limits_{\alpha  = 1}^n {{x_{\alpha i}}x_{\alpha j}^{}{x_{\alpha k}}x_{\alpha l}^{}} } \right] + \sum\limits_{\alpha  \ne \omega }^n {E\left[ {{x_{\alpha i}}x_{\alpha j}^{}} \right]E\left[ {{x_{\omega k}}x_{\omega l}^{}} \right]}  - {n^2}{\sigma _{ij}}{\sigma _{kl}} = \\
\sum\limits_{\alpha  = 1}^n {E\left[ {{x_{\alpha i}}x_{\alpha j}^{}{x_{\alpha k}}x_{\alpha l}^{}} \right]}  + n\left( {n - 1} \right){\sigma _{ij}}{\sigma _{kl}} - {n^2}{\sigma _{ij}}{\sigma _{kl}} = \\
n\left( {{\sigma _{ij}}{\sigma _{kl}} + {\sigma _{il}}{\sigma _{jk}} + {\sigma _{ik}}{\sigma _{jl}}} \right) + n\left( {n - 1} \right){\sigma _{ij}}{\sigma _{kl}} - {n^2}{\sigma _{ij}}{\sigma _{kl}} = n\left( {{\sigma _{il}}{\sigma _{jk}} + {\sigma _{ik}}{\sigma _{jl}}} \right)
\end{array}
The moments of $x_1,...,x_p$ with a joint normal distribution can be obtained from the characteristics function. (The first four moments of $x$ are available in Anderson (2003) An Introduction to Multivariate Statistical Analysis. 3ed.)
