What is the cubic expectation (third-order moment) of a complex gaussian vector (say, E[$aa^{T}a$])? Note: I also posted this question on MATHEMATICS.
For a real gaussian vector, an explicit formula for the cubic expectation can be found in Matrix Reference Manual (search 'Cubic Expectations' in this link for detail), i.e.,
say $a$ is a complex gaussian (vector) process with the mean $E[a] = m$ and covariance $Cov[a] = S$, then we have
E[$aa^{T}a$] = E$[a^{T}aa^{T}]^{T}$ = $2Sm + (tr(S)+ m^{T}m)×m$
where $×$ stands for the operation of skew-symmetric matrix (search 'skew-symmetric matrix' in this link for detail).
But what about the cubic expectation of complex gaussian vectors?
In Matrix Reference Manual, you can find explicit formulae for even-order moments of complex vectors, but not for odd-order moments. Usually, we talk about zero-mean vectors, therefore the cubic expectation is ignored. However, for conditional expectation, the odd-order moments may not be zero.
Does anyone know how to find an explicit formula like the one posted above? Any thought would be appreciated.

If you write a = m + x where m is the mean and x has zero mean then you can expand your cubic expression by multiplying it out. All the odd powers of x in your expansion will have zero mean and you can write down expressions for the even powers. Hope this helps.
