I have been stacked for a while with the following problem: Consider two samples of iid observations $X^1=\{X_1^1,\dots,X_n^1\}$ and $Y_1=\{Y_1^1,\dots,Y_n^1\}$ where $X_i^1 \sim \mathcal{N}(0,1)$ and $Y_i^1 \sim \mathcal{N}(0,1)$ and let $\rho_1$ be their sample correlation. Now consider a given $n \times n$ stochastic matrix $W$, let $X^2=WX^1$, $Y^2=WY^1$ and $\rho_2$ be the sample correlation between the vectors $X^2$ and $Y^2$ (Notice that $X_i^2$'s do not follow the same distribution anymore, since each variance depends on the respective weights in $W$).
My first question is whether we can determine the distributions of $\rho_1$ and $\rho_2$? I know that they both have mean equal to zero. In fact, both of them have symmetric pdfs, so all their odd moments are zero (including the mean). And I have also shown that the even moments of $\rho_2$ are larger than of $\rho_1$.
If we cannot find the distributions, could we at least compare $E(|\rho_1|)$ and $E(|\rho_2|)$. Finally, can we generalise this relationship for any $X^t$ and $Y^t$ as long as we know the distributions of $X^{t-1}$ and $Y^{t-1}$ and $W$?
I hope the question is clear. Any help would be more than welcome. Thanks in advance.