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Let $x_1$, $x_2$,.. $x_n$ be independent and identically distributed draws from a p-dimensional multivariate normal distribution, i.e., $x_i \sim N_p(0, A)$ (A is the co-variance matrix) and they form a $p\times n$ data matrix $X=[x_1 \;\; x_2 \;\; .. x_n]$. In that case, the distribution of a p × p random matrix $M=XX^T$ is said to have the Wishart distribution.

Can you kindly tell me, if there is any distribution function available for $XX^T$ when $X=[x_1 \;\; x_2 \;\; .. x_n]$, where each $x_i \sim N_p(0, A_i)$, i.e., they are independent but not identically distributed. Thank you very much.

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  • $\begingroup$ Are the $A_i$ fixed and known? $\endgroup$ – wrvb Jun 21 '17 at 3:29
  • $\begingroup$ yes, $A_i$s are fixed and known. Any help? $\endgroup$ – Sabyasachi G Jun 21 '17 at 11:42
  • $\begingroup$ After thinking about it, I believe the answer is no, there is no simple distribution for $XX^T$. If the $A_i$ are scalar multiples of each other ($A_i=a_i A$ for scalar $a_i$) then $XX^T$ is Wishart distributed, but that doesn't help in general. $\endgroup$ – wrvb Jun 22 '17 at 19:57
  • $\begingroup$ Yup, thanks anyways. $\endgroup$ – Sabyasachi G Jun 25 '17 at 14:57
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One has $M = \sum_{i=1}^n x_i x_i^T$ and $x_i x_i^T \sim W_p(1, A_i)$.

Thus, you are asking for the distribution of the sum of $n$ independent $W_p(\nu, A_i)$ random matrices. This distribution is not simple. For $n=2$, its density is given in Problem 3.5 of Gupta & Nagar's book Matrix variate distributions. This density involves an hypergeometric function of matrix argument.

Note that in the case $p=1$, this is the distribution of a linear combination of independent $\chi^2$ distributions, which is already not easy.

More generally, the density of the sum of $n$ independent noncentral Wishart distributions (i.e. when the $x_i$ are not centered) has been derived by Chikuse & Davis in Some properties of invariant polynomials with matrix arguments and their applications in econometrics (1986). This density is given by a series involving generalized Laguerre polynomials.

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