# Find the limiting distribution of Sum over Sum of Squares

Having a little trouble with this one:

Suppose $X_1, X_2, \ldots$ are iid standard normal random variables. Let $W_n = \sqrt{n} \frac{X_1 + \cdots + X_n}{X_1^2 + \cdots + X_n^2}$. Find the limiting distribution of $W_n$ as $n \to \infty$.

Too bad convergence in distribution isn't closed under division. Can't get slutsky's to apply.

Notice that $\frac{X_1 + X_2 + \ldots+ X_n}{\sqrt n}$ is a a standard gaussian variable. Then apply Slutsky's theorem.
• $\frac{\sum (x_i^2)}{n}$ converges in probability by WLLN? Jan 3, 2013 at 13:45
The numerator is normally distributed with mean $0$ and variance $n$, and the denominator is distributed $\chi^2$ with $n$ degrees of freedom. Start there.
We can find the result with law of large numbers and central limit theorem, writing $$W_n=\color{blue}{\frac n{\sum_{j=1}^nX_j^2}}\cdot\color{red}{\frac 1{\sqrt n}\sum_{j=1}^nX_j}.$$ The blue term converges to $E(X_1^2)^{-1}$ in probability by the law of large numbers.