Consider a random vector $\boldsymbol{x} \in \mathbb{R}^N$ and the identity matrix $\boldsymbol{I}_N \in \mathbb{R}^{N\times N}$. I have to compute the expected value of the following Kronecker product, \begin{equation} \mathbb{E} \Big[ (\boldsymbol{x}^\top \otimes \boldsymbol{I}_N) \otimes (\boldsymbol{x}^\top \otimes \boldsymbol{I}_N) \Big]. \end{equation}

I know all the moments of the random vector $\boldsymbol{x}$, for example, iI have the closed form of $\mathbb{E}[\boldsymbol{x}]$, $\mathbb{E}[\boldsymbol{x} \otimes \boldsymbol{x}]$ and of course $\mathbb{V}[\boldsymbol{x}]$.

I was looking for some nice property that can allow me to explicit in only the random vector from the above equation, in order to use the closed formula that I have mentioned. I'm trying so hard with different vectorization approaches but I haven't find a solution yet. Any comments or suggestions are always welcome! Thank you in advance!

  • $\begingroup$ Clearly the expression is a quadratic function of $x,$ so this is just a matter of keeping track of the indices. $\endgroup$ – whuber Sep 4 '18 at 13:58

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