Showing convergence in distribution example

I took my first probability theory course last semester and I've been trying to work through problems from old qualifying exams for practice. I haven't quite been able to figure out problems like the following.

Suppose $X_1, \ldots , X_n$ are independent and uniformly distributed on $[0,1]$, and let $$U_n = \sqrt{n} \frac{X_1 + \ldots + X_n}{X_1^2 + \ldots + X_n^2}.$$ Show that $U_n$ converges in distribution and find its limit.

I initially thought to write $U_n$ as $$U_n=\sqrt{n} \frac{\frac{1}{n}(X_1 + \ldots + X_n)}{\frac{1}{n}(X_1^2 + \ldots + X_n^2)}$$ because we have $$\frac{1}{n}(X_1 + \ldots + X_n) \xrightarrow{P} \mathbb{E}[X_1]=0$$ $$\frac{1}{n}(X_1^2 + \ldots + X_n^2) \xrightarrow{P} \mathbb{E}[X_1^2]=\frac{1}{3}$$ by the weak law of large numbers.

Also, if we didn't have the term in the denominator, we could conclude that $$\sqrt{n} \frac{1}{n}\big(X_1 + \ldots + X_n\big) \xrightarrow{D} \mathcal{N}\Big(0,Var(X_1)\Big)$$ by the Central Limit Theorem.

Those are the main things that have occurred to me so far. I would greatly appreciate any suggestions (or corrections). Thanks in advance!

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1 Answer

Write $$U_n:=\color{green}{\frac{X_1+\cdots+ X_n}{\sqrt n}}\cdot\color{red}{\frac n{X_1^2+\dots+X_n^2}},$$ then use and show the following:

If $Y_n\to Y$ in distribution and $Z_n\to c\neq 0$ in probability, then $Y_nZ_n\to cY$ in distribution.

Condlue

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Nice! Thanks for the help. – eeeeeeeeee Jan 4 '13 at 18:24
I tried to add the missing plus sign in your answer (no pun intended), but apparently edits must be at least 6 characters. Oh well :) – eeeeeeeeee Jan 4 '13 at 18:27
Just thought I'd mention that Slutsky's theorem has proved to be pretty useful! I've used it in several problems since your suggestion. – eeeeeeeeee Jan 5 '13 at 22:45