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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?

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  • $\begingroup$ Are you sure the variables are dependent? $\endgroup$ – ekvall Jun 11 '16 at 9:37
  • $\begingroup$ @Student001 Are they not? I thought both were functions of $X_i$'s so they must be..(Forgive me if I am wrong.. I am an undergrad and my probability theory foundations are nit very strong) $\endgroup$ – Qwerty Jun 11 '16 at 9:41
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    $\begingroup$ I think, without having gone through the details, that you can use Basu's theorem and the iid assumption to argue that $V_n$ is independent of $U_{n/2}$ and $U_{n} - U_{n/2}$ and thus also of $U_n$. I can't guarantee it will work, but that was my first hunch so it may be worth a try. $\endgroup$ – ekvall Jun 11 '16 at 9:47
  • $\begingroup$ @Student001 I don't know Basu's theorem. Can you please explain it and show how it can be used to prove the independence? Wiki has explained it badly. So if you explain it as an answer then it might be of great help to me as an example.. $\endgroup$ – Qwerty Jun 11 '16 at 9:52
  • $\begingroup$ Ok. Two more questions: do you know Slutsky's theorem, and is this for coursework or anything else that falls under the self-study tag? $\endgroup$ – ekvall Jun 11 '16 at 10:18
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Although the brute force method I mentioned in the comments may work, there is an easier way which does not rely on Basu's theorem, and it also avoids integration of the joint density of transformed random variables. I leave some details out because the question is of homework-style.

Write $$ Z_n = \frac{\frac{1}{n}U_n}{\frac{1}{n}V_n}. $$

  1. Find the distribution of the numerator random variable using a density transform and consider how it depends on $n$
  2. Note that all moments of $X_i$ exist and are finite. In particular, the variance of $X_i^2$ is finite. Thus, you can use Chebyshev's inequality to prove that $V_n/n \to 1$ in probability
  3. Prove that, in general, if for some sequence of random variables $W_n$ it holds that $W_n \overset{p}{\to} 1$, then $Y/W_n \overset{d}{\to}Y$. Note that you do not need independence for this.
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  • $\begingroup$ Well I didn't get your 2nd point. Can you please explain more? $\endgroup$ – Qwerty Jun 11 '16 at 12:24
  • $\begingroup$ @Qwerty do you know any standard convergence theorems, like the central limit theorem and law of large numbers? Please also add the self-study tag to your question, it seems appropriate. $\endgroup$ – ekvall Jun 11 '16 at 12:34
  • $\begingroup$ I added the self study tag. To speak the truth, the question is from a book exercise, but unfortunately, the problem is given in an exercise before the chapter on central limit theorem and LLN s so I have not studied those. However I have studied the convergence types, like weak convergence, convergence in probability, almost sure convergence, and $r$'th mean convergence.. $\endgroup$ – Qwerty Jun 11 '16 at 12:47
  • $\begingroup$ @Qwerty I changed the hints to not rely on any of those theorems. $\endgroup$ – ekvall Jun 11 '16 at 15:00

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