# 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.

• Your opening notation/condition is unclear, for me. $x_\alpha$ seems to be a multivariate vector. Is it to come from the specified normal distribution, or is it a sample realization (number of cases, i.e. many such vectors collected) giving exactly such distribution? Also, I didn't quite understand your definition of matrix A. Is it a vector or matrix? Apr 21, 2017 at 9:06
• @ttnphns: $x_\alpha$ is one sample realization vector. Matrix $A$ is the scatter matrix over the sample of $n$ such vectors (note that the sum is going from $\alpha=1$ to $n$). Apr 21, 2017 at 9:34
• Note that $A/n$ is a sample covariance matrix, so one can equivalently ask about the covariance between two elements of the sample covariance matrix. I added [covariance-matrix] tag. Apr 21, 2017 at 9:37
• You need to use formulas for the fourth moments of Normal variables. It's just a matter of working with the (four) indexes in $E[a_{ij}a_{kl}]$, unpacking them in terms of the $x$'s, and paying attention to when two, three, or four of the same $x$ component appear together.
– whuber
Apr 21, 2017 at 13:51

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.)