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(Trying to solve the same underlying problem as this.)

Let's say we know the marginal probability density functions $p_i(x)$ of a set of zero-mean random variables $\{X_i\}_{i=1}^N$ as well as the second-order statistics $C_{ij} = \mathbb E [X_i X_j^*]$. Assume that $X_i$ are complex-valued circularly-symmetric random variables (i.e., real and imaginary parts are independent).

Is there a principled way of producing an estimate of the joint distribution $p(x_1, x_2, \ldots, x_n)$?

In the case where the marginals are unknown, the maximum entropy distribution is simply the multivariate normal for both the real and imaginary parts. Is there a similar process that allows one to account for the additional information provided by the marginals?

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