The question is pretty much in the title,
I need to find an approximate distribution of $XY$ when $(X,Y)$ follow a Bivariate Normal Distribution where $X$ and $Y$ are each $N(0,1)$ distributed and $Cov(X,Y) = \rho.$
I proceeded to find the required using transformation of variables, but I am stuck at the following integration,
$\displaystyle \int_{-\infty}^{\infty} \left| \frac{1}{u}\right| \text{exp} \left\{ \frac{-1}{2(1-\rho^2)}\left(u^2 + \frac{v^2}{u^2}\right)\right\}\,du$
How do I proceed to solve this integral? Or is there any other way obtaining the required distribution?
This problem is however an intermediate step of another question that I am trying to solve,
Suppose ${(X_1, Y_1), . . . ,(X_n, Y_n)}$ is a random sample from a bivariate normal distribution with $E(X_i) = E(Y_i) = 0, \, Var(X_i) = Var(Y_i) = 1$ and unknown $Corr(X_i, Y_i) = \rho \in (−1, 1),$ for all $i = 1, . . . , n.$ Define $W_n = \frac{1}{n}\sum_{i = 1}^n X_iY_i$. For large $n$, obtain an approximate level $(1 − \alpha)$ two-sided confidence interval for $\rho$, where $0 < \alpha < 1.$
Though, I know about the Fisher's z transform, and its asymptotic distribution. But I think the question requires me to give an answer in terms of $W_n$ only. How should I solve this?
Can I use CLT in this? Because I ran a simulation for 1000 times and the distribution of $W_n$ came out to be convincingly normal.