# Variance of sum of square (not independent) random vectors

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).

• 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? – whuber Apr 3 '17 at 20:35
• 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 – Mr.Al Apr 4 '17 at 8:17
• 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.) – whuber Apr 4 '17 at 15:28
• 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 $S$. – Mr.Al Apr 4 '17 at 17:38