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I have that $\mathbf{x}_{i}=(x_{i1},\ldots,x_{ip})' \sim N_{p}(0,V)$ and I'm interesting in the variance of:

$$ S = \sum_{i=1}^{n} \mathbf{x}_{i}\mathbf{x}_{i}' $$

for the case when the vectors are correlated. In my case: $\operatorname{Cov}(\mathbf{x}_{i},\mathbf{x}_{j})= - \frac{V}{n-1}$. I now that:

$$ \operatorname{Var}(\mbox{vec} \ \mathbf{x}_{i}\mathbf{x}_{i}' )=\operatorname{ Var}(\mathbf{x}_{i} \otimes \mathbf{x}_{i}) = (I-K_{p})(V \otimes V)$$

and

$$ \mbox{Cov}(\mbox{vec} \ x_{i}x_{i}, \mbox{vec} \ x_{j}x_{j}) = \operatorname{Var}(\mathbf{x}_{i} \otimes \mathbf{x}_{j}' ) = V \otimes V + K_{p}(\operatorname{Cov}(x_{i},x_{j}) \otimes \operatorname{Cov}(x_{i},x_{j}))$$

where $K$ is the commutation matrix. Both results are by Magnus and Neudecker (1979). I know that in the case of independence $Cov(\mathbf{x}_{i},\mathbf{x}_{j}) = 0$ (Well know result):

$$ \operatorname{Var}(S) = n(I-K_{p})(V \otimes V) $$

Moreover, $S$ is Wishart distributed. But the extension for non-independent vectors doesn't seem to be right to me:

$$ \operatorname{Var}(S) = \sum_{i=1}^{n} \operatorname{Var}(\mathbf{x}_{i} \otimes \mathbf{x}_{i}) + \sum_{i=1}^{n} \sum_{j \neq i} \operatorname{Var}(\mathbf{x}_{i} \otimes \mathbf{x}_{j})$$

Can you help me with the last expression? (it's maybe wrong).

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    $\begingroup$ Your overloading of the symbol "$V$" to mean (apparently) "variance" as well as the covariance matrix $V$ makes this question very confusing. Could you clear up this ambiguity? $\endgroup$ – whuber Apr 3 '17 at 20:35
  • $\begingroup$ Thanks, I made some corrections. I referring the variance of $x_{i}x_{i}'$ as $V(x_{i} \otimes x_{i})$ and the covariance of $x_{i}$ and $x_{j}$ as $V(x_{i} \otimes x_{j})$, such as Magnus and Neudecker (1979). Hope this clear up the ambiguity $\endgroup$ – Mr.Al Apr 4 '17 at 8:17
  • $\begingroup$ It's just as confusing as ever because of your double use of the symbol "$V$". For instance, where you write "$V\otimes V$" do you mean the tensor product of a matrix $V$ with itself or the tensor product of the variance with itself? Where you write "$V(\mathbf{x}_{i} \otimes \mathbf{x}_{j})$" does that mean the variance of a tensor product or the product of $V$ with a matrix formed with a tensor product? Indeed, what is your question? Do you want to find the variance of $S$ or do you want to find the variance of $V(S)$?? (The latter is literally what you ask.) $\endgroup$ – whuber Apr 4 '17 at 15:28
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    $\begingroup$ You're right I was using double use of $V$. $V$ is the variance matrix of $x_{i}$. I corrected it. I want to find the variance of $S$. $\endgroup$ – Mr.Al Apr 4 '17 at 17:38

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