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Let $X_1,X_2,...$ be iid RVs with mean $0$, variance $1$, and $EX_i^4\lt\infty$. Find the limiting distribution of

$$Z_n=\sqrt{n}\frac{X_1X_2+X_3X_4+...+X_{2n-1}X_{2n}}{X_1^2X_2^2+...+X_{2n}^2}$$

The only theorem I have ever seen talk about the 4th moment is: If $X_1,X_2,...$ are iid RVs with common mean $\mu$ and finite fourth moment, then

$$P[\lim_{n\to\infty}(\rm{sample\, mean})=\mu]=1$$

I was also thinking about the CLT. Intuition tells me the answer should be standard normal, but I'm not sure how to get to that (if I'm right).

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    Hint: What is the limit in distribution of $$\frac{X_1X_2+X_3X_4+...+X_{2n-1}X_{2n}}{\sqrt{n}}\ ?$$ What is the almost sure limit of $$\frac{X_1^2X_2^2+...+X_{2n-1}^2X_{2n}^2}n\ ?$$ Ergo? – 2017-01-22
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    That second limit I believe is 1 by the Strong Law of Large Numbers. The first I am not sure. – 2017-01-22
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    I believe it should be standard normal by the CLT, but I don't know how to apply it with the multiplied terms. – 2017-01-23
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    Well, this is $$\frac{Y_1+\cdots+Y_n}{\sqrt{n}}$$ with $$Y_n=X_{2n-1}X_{2n}$$ so what is the problem? – 2017-01-23
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    @Did Does the $2n$ change anything? Like in those two denominators why don't we change $\sqrt{n}$ and $n$ to $\sqrt{2n}$ and $2n$? – 2017-01-23
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    The $X_i$'s are not necessarily normal in case you didn't see that. – 2017-01-23
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    Not normal: yes, not needed, not used. – 2017-01-23

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