# Sum of Wishart matrices

Considering two matrices, $H_1$ and $H_2$, that are independent of each other and follows complex wishart distributions as $\mathcal{CW} _m(n_1,\Sigma_1)$ and $\mathcal{CW} _m(n_2,\Sigma_2)$ respectively. Now considering the sum

$$Z=H_1+H_2.$$

Will $Z$ also follow a Wishart distribution?

I have only been able to find one example of something like this, in this paper, where the two matrices only differ in degrees of freedom. Is there a similar expression for when the degrees of freedom, $n$, but also the covariance matrix, $\Sigma$, differ?

• You haven't said anything about the JOINT distribution of $H_1$ and $H_2$. The result I know about the sum of Wishart matrices is that if $H_1\sim\mathcal{CW} _m(n_1,\Sigma)$ and $H_2\sim\mathcal{CW} _m(n_2,\Sigma)$ AND $H_1$ and $H_2$ are independent, then $H_1+H_2\sim\mathcal{CW} _m(n_1+n_2,\Sigma)$. I don't know any results that don't assume independence. Of course, that doesn't mean there aren't any, but I'd still expect a hypothesis saying something about their JOINT distribution. Nov 11, 2013 at 16:16
• Thanks for pointing this out. It is edited. Nov 11, 2013 at 21:20
• If they are independent and $\Sigma_1=\Sigma_2$ then certainly the sum has a Wishart distribution with $n_1+n_2$ degrees of freedom. ${}\qquad{}$ Sep 17, 2015 at 23:54