Bounty: 50
I’m looking for the properties of a degenerate complex normal distribution. Wikipedia has a section on degeneracy in the real multivariate normal case here, stating that the density (in a subspace of $mathbb R^k$ where the distribution is supported) looks like:
$$f(mathbf{x}) = (text{det}^{*}(2piSigma))^{-1/2} e^{-1/2 (x-mu)^TSigma^+(x-mu)}$$
… where $text{det}^*$ is a pseudo-determinant (product of non-zero eigenvalues) and $Sigma^+$ is the pseudo-inverse.
I have a couple of questions relating to whether or not such a formulation exists for the complex case.
Question 1: If $Z sim mathcal{C}N(mu, Gamma, C)$ where the support is $mathbb C^2$ (for example), then multiplication of $Z$ with a matrix $Ain mathbb{C}_{3,2}$ would yeild: $AZ sim mathcal{C}N(Amu, AGamma A^H, ACA^T)$ where $AZ in mathbb{C}^3$ and could, in general, be degenerate.
Assuming that $A$ is unknown, how would one find the subspace of $mathbb{C}^{3}$ where the density would have a support, and in this subspace, what would the density be? Would be be similar to the real case wherein the determinants and inverses are replaced by their pseudo-counterparts?
Question 2: On the wikipedia article, it is shown that the density of a complex normal can be written as:
$$f(z) = tfrac{sqrt{detleft(overline{P^{-1}}-R^{ast} P^{-1}Rright)det(P^{-1})}}{pi^n},
e^{ -(z-mu)^astoverline{P^{-1}}(z-mu) +
operatorname{Re}left((z-mu)^intercal R^intercaloverline{P^{-1}}(z-mu)right)}$$
…where $P, R$ are functions of $Gamma, C$. I assume that this density is only valid if, for example, $Gamma$ is invertible. In a specific problem setting, I’ve encountered a likelihood that can be written in this form, but $detleft(overline{P^{-1}}-R^{ast} P^{-1}Rright) = 0$. Does this imply that the density is degenerate? Or is it the case that my likelihood isn’t a likelihood at all?
