Suppose I have an $n$-dimensional complex, zero mean normal distribution with covariance matrix $\Sigma$, which is not diagonal. Denoting each of the random variables as $x_1, \dots ,x_n$ I would like to find the $n$ dimensional PDF for a new set of variables $z_i = x_ix_i^* x_1 x_1^* \,\,\, \forall \,\, i \in \{1,\dots,n\} $. After reading around the topic of the algebra of random variables I'm still unsure whether techniques to perform this sort of task exist. I've read parts of M.D. Springer's book on the topic but there doesn't seem to be an example where a multivariate PDF is derived for products of dependent variables.

What I'd really like to know is if techniques exist to find the PDF. Also, any help finding the relevant literature (if it exists) would be helpful.

Would the answer be $$ p_{Z_1,\dots,Z_n}(z_1,\dots,z_n) = \int_{-\infty}^{\infty} \mathrm{d}^2 x_1 \dots \int_{-\infty}^{\infty} \mathrm{d}^2 x_n \,\,\,p_{X_1,\dots,X_n} (x_1,\dots,x_n) \prod_{i=1}^n \delta(z_i-|x_1|^2|x_i|^2) $$ where $p_{Z_1,\dots,Z_n}(z_1,\dots,z_n)$ is the $n$-dimensional PDF of the variables $z_i,\dots,z_n$ and $p_{X_1,\dots,X_n} (x_1,\dots,x_n)$ is the $n$-dimensional PDF of the complex variables $x_1,\dots,x_n$? I've completely guessed at this formula from various solutions I've found for a single random variable $z$ defined as $z=xy$.


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