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Let $\mathbf{x}$ and $\mathbf{y}$ be two independent, complex, $N$-dimensional random vectors with zero mean and covariance matrices $\boldsymbol{\Sigma}_{x}$ and $\boldsymbol{\Sigma}_{y}$. Let us know define the complex random vector $\mathbf{z}$ as \begin{equation} \mathbf{z} = \mathbf{x} + \mathbf{y}, \end{equation} such that $\mathbf{z}$ is a complex elliptically symmetric - distributed random vector with mean zero and scatter matrix given by $\boldsymbol{\Sigma}_{z} = \boldsymbol{\Sigma}_{x} + \boldsymbol{\Sigma}_{y}$.

Which distributions could $\mathbf{x}$ and $\mathbf{y}$ have to satisfy this requirement on the distribution of $\mathbf{z}$?

Thanks!

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