Elliptical random vector as sum of two vectors

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 $$$$\mathbf{z} = \mathbf{x} + \mathbf{y},$$$$ such that $$\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!