Well , I have the following problem..
Let $X_1,\cdots ,X_{2n}$ be iid $N(0,1)$ random variables. Define $$U_n=\left({X_1\over X_2}+{X_3\over X_4}+\cdots +{X_{2n-1}\over X_{2n}}\right)$$ $$V_n=X_1^2+X_2^2+\cdots+X_n^2$$ $$Z_n={U_n\over V_n}$$ Find limiting distribution of $Z_n$
My work:
I have found out that $$f_{_{U_n}}(u)={n\over \pi(n^2+u^2)}$$ and $$V_n\sim \chi^2(n)$$
Now I am stuck . I know that to find $Z_n$ I first need the joint $pdf$ of $U_n\ \&\ V_n$ and then apply convolution formula. But I am not being able to find the joint $pdf$. I know that had $U_n\ \&\ V_n$ been independent, it would have been just the product of the two densities. But how to do that when the variables are dependent?