# Distribution of the sum of Wishart distributed random matrices

Suppose

$$$$A_1 \sim \mathcal{W}(A_1; \Psi_1, v_1)\\ A_2 \sim \mathcal{W}(A_2; \Psi_2, v_2),$$$$

where $$\mathcal{W}$$ is the Wishart distribution and $$A_i$$, and $$\Psi_i$$ are PSD matrices. Does

$$$$A_1 + A_2$$$$ have a closed-form distribution?

• It does have a closed form (Wishart) only if the scale matrices coincide. Commented Jan 9 at 5:31
• Hey do you mean the matrices commute? ie $\Psi_1\Psi_2 = \Psi_2\Psi_1$ ? I cant find a definition of coinciding matrices. tx Commented Jan 9 at 6:01
• I mean that they should be equal. Commented Jan 9 at 6:03

As per my comment, if $$\Psi_1=\Psi_2$$, then it can be shown that $$A = A_1+A_2$$ follows a Wishart distribution with scale matrix $$\Psi_1$$ and degrees of freedom $$\nu_1+\nu_1$$.
On the other, if $$\Psi_1\neq \Psi_2$$ (but they are both square and with $$p$$ rows), the distribution is not Whishart anymore. Indeed, in this case, the pdf of $$A$$ is
$$\left\{2^{\frac{1}{2}(\nu_1+\nu_2)p}\Gamma_p\left(\frac{1}{2}(\nu_1+\nu_2)\right)\det(\Psi_1)^{\frac{1}{2}\nu_1}\det(\Psi_2)^{\frac{1}{2}\nu_2}\right\}^{-1}\text{etr}\left(-\frac{1}{2}\Psi_1^{-1}A\right)\det(A)^{\frac{1}{2}(\nu_1+\nu_2-p-1)} {_1}F_1\left(\frac{1}{2}\nu_2;\frac{1}{2}(\nu_1+\nu_2);\frac{1}{2}(\Psi_1^{-1}-\Psi_2^{-1})A\right), A>0,$$
where $$\Gamma_p(a)= \int_{A>0}\text{etr}(-A)\det(A)^{a-\frac{1}{3}(p+1)}dA$$, where $$\text{Re}(a)>\frac{1}{2}(p-1)$$ and the integral is over the space of $$p\times p$$ symmetric positive definite matrices, $${_1}F_1$$ is the confluent hypergeometric function (of the first kind) and $$\text{etr}(A) = \exp(\text{tr}A)$$.