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Is there any result showing that a sum of independent Wishart with same degrees of freedom but different scale matrices is a Wishart?

For example, if I have two random variables: $$ Y \sim W_p(n,\sigma_1)\ \\ X \sim W_p(n, \sigma_2), $$

where $p$ is the dimension of matrices, will $Y+X$ be equal to $W_p(n, \sigma_1 + \sigma_2)$?

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    $\begingroup$ When $p=1$, Wishart is Gamma, and the sum is not Gamma. $\endgroup$ – Stéphane Laurent Feb 25 '14 at 13:47
  • $\begingroup$ Yes, that's right. Even with the same shape parameters, sum of Gammas is not Gamma. Thank You, Stéphane. – $\endgroup$ – Igor Souza Feb 25 '14 at 14:26

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