Showing convergence of a random variable in distribution to a standard normal random variable

Question

Let $X_1$, $X_2$, $X_3$,$\ldots$ be independent and identically distributed random standard normal random variables.

Define the following: $$Z_n = \sqrt{n} \frac{X_1 + X_2 + \cdots + X_n}{X_1^2 + X_2^2 + \cdots + X_n^2}$$

I am trying to prove that $Z_n$ converges in distribution to a standard normal random variable $Z$.

This looks tantalizingly close to something to do with the Central Limit Theorem, but I cannot quite see how to make the CLT apply here.

I have attempted to show this convergence by use of moment generating functions, but that got very messy very quickly.

Is there a simple way to show the desired result using the CLT, or is it more involved?

Answer 1

Rewrite $Z_n$ as $$Z_n = \frac{X_1 +...+X_n}{\sqrt{n}} \cdot \frac{n}{X_1^2 + ... + X_n^2}= A_n \cdot B_n.$$ Now, $A_n$ follows a standard normal and $B_n \to 1$ in probability by the weak law of large number and the continuous mapping theorem (since $\Bbb E[X_1^2]$).

Therefore, by Slutsky's theorem $Z_n$ converges in distribution to a standard normal.