Fourth Moment in Terms of Correlations

[marcusl]

For multivariate normal distributions, Isserlis' theorem gives us moments in terms of cross-correlations, e.g.,

$$\operatorname{E}[\,x_1x_2x_3x_4\,] = \operatorname{E}[x_1x_2]\,\operatorname{E}[x_3x_4] + \operatorname{E}[x_1x_3]\,\operatorname{E}[x_2x_4] + \operatorname{E}[x_1x_4]\,\operatorname{E}[x_2x_3] = r_{12}r_{34}+r_{13}r_{24}+r_{14}r_{23}$$

Does this equation hold generally for non-normal distributions?
And does it change for complex (rather than real) quantities?
I am trying to analyze the complex signals received by an antenna array.

Thank you!

 

[marcusl]

I think I've figured out the answer to my questions. The fourth-order cumulant (FOC) is given by

$$\operatorname{cum}[\,x_1x_2x_3x_4\,] = \operatorname{E}[\,x_1x_2x_3x_4\,] - \operatorname{E}[x_1x_2]\,\operatorname{E}[x_3x_4] - \operatorname{E}[x_1x_3]\,\operatorname{E}[x_2x_4] - \operatorname{E}[x_1x_4]\,\operatorname{E}[x_2x_3]$$

Note that the first term on the right is the multivariable fourth moment, while the remaining terms are the fourth moment of normally distributed rv's by Isserlis' theorem. We add the following properties of cumulants: the FOC for general random variables is, in general, non-zero, while the FOC for normally distributed random variables is identically zero. Putting these all together, then the expression in my first email must be specific to normally distributed rv's.