Variance of the sum of elements of a Wishart distributed matrix

Question

Looking for the variance of $S=\sigma _{1,3}-\sigma _{1,4}-\sigma _{2,3}+\sigma _{2,4}$, where $\sigma_{i,j}$ are Wishart-distributed elements of the random matrix

$$\Sigma =\left( \begin{array}{cccc} \sigma _1^2 & \sigma _{1,2} & \sigma _{1,3} & \sigma _{1,4} \\ \sigma _{1,2} & \sigma _2^2 & \sigma _{2,3} & \sigma _{2,4} \\ \sigma _{1,3} & \sigma _{2,3} & \sigma _3^2 & \sigma _{3,4} \\ \sigma _{1,4} & \sigma _{2,4} & \sigma _{3,4} & \sigma _4^2 \\ \end{array} \right)$$ the $m$-sample estimation of the covariance matrix of 4 multivariate Gaussian distributed random variables with $n$ observations each.

(I tried Math Stack Exchange with no result).

Answer 1

Let $a = (\begin{matrix} 1 & -1 & -1 &1\end{matrix})$ and $\sigma = (\begin{matrix} \sigma_{1,3} & \sigma_{1,4} &\sigma_{2,3} & \sigma_{2,4}\end{matrix})'$.

Then $S=a\sigma$. Let $V$ be the variance-covariance matrix of $\sigma$. We have $$\mathrm{Var}(S) = aVa'$$

About $V$, the variance-covariance matrix of 4 multivariate Gaussian distributed random variables is needed. Then referring to eqn (1) on page 4 of http://www.math.unm.edu/~fletcher/Wishart.pdf, you can get the answer.