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Let $X_1,X_2,...,X_{2n}$ be iid $N(0,1)$ RVs. Define

$U_n=(\frac{X_1}{X_2}+\frac{X_3}{X_4}+...+\frac{X_{2n-1}}{X_{2n}})$,

$V_n=X_1^2+X_2^2+...+X_n^2$,

and $Z_n=\frac{U_n}{V_n}$.

Find the limiting distribution of $Z_n$.

My thoughts: $V_n$ is the sum of squares of standard normal RVs, so it should be chi square with $n$ degrees of freedom as its parameter. The correct result should be that $Z_n$ follows a Cauchy($1,0$) distribution. I don't know what $U_n$ is or how Cauchy is supposed to come from chi square.

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    Recall that if $X,Y$ are $N(0,1)$ then $X/Y$ is Cauchy. Also, the sum of Cauchy is again Cauchy (with a different parameter). – 2017-01-21
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    Ah that makes more sense. Then I just need to try and prove that $V_n$ converges in probability to 1? – 2017-01-21
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    $U_n/n$ is a Cauchy distributed r.v. and $V_n/n\to 1$ a.s. by SLLN, so the limiting distribution of $Z_n$ is the Cauchy distribution. – 2017-01-21

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