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Say I have 4 random variables. $X^{(1)}$ and $X^{(2)}$ are jointly multivariate normal with mean 0 and covariance $\Sigma_X$, and $Y^{(1)}$ and $Y^{(2)}$ are jointly multivariate normal with mean 0 and covariance $\Sigma_Y$. There are no dependencies between these two pairs. Now I want to know the following expected value: $$E\left [\frac{\mathrm{cov}(X_{1:N}^{(1)},X_{1:N}^{(2)})}{\sqrt{\mathrm{var}(X_{1:N}^{(1)}+Y_{1:N}^{(1)})}\sqrt{\mathrm{var}(X_{1:N}^{(2)}+Y_{1:N}^{(2)})}}\right ] $$

If it helps, I think I'm right in saying that this fraction is essentially a semi-partial correlation. That is, if you define $Z^{(1)}=X^{(1)}+Y^{(1)}$ and $Z^{(2)}=X^{(2)}+Y^{(2)}$, it is the semi-partial correlation between $Z^{(1)}$ and $Z^{(2)}$, controlling for $Y^{(1)}$ and $Y^{(2)}$. I don't particularly care about that interpretation though, I'm really just interested in finding the solution to this expected value.

So far I've managed to realize that the numerator and denominator aren't independent, and thus the problem can't be easily partitioned into independent expected values. It seems, then, that it would have to be worked out in terms of the joint distribution of the numerator and denominator, which is rather daunting. So I guess my hope is someone here is equal to the task, has some pointers, or maybe even knows about an established solution?

From doing some simulations, it seems that the distribution of this thing looks just like that of a correlation coefficient, so I have a suspicion that the pdf will have the same form, with different parameters (since the denominator here is different from what it would be in a normal correlation). But then the question is what those parameters would be.

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