convergence in distribution: show limit is standard normal

Question

$(Y_n)_{n\geq 1}$ is iid standard normal. Then how can I prove the following:

$$\frac{Y_1}{(n^{-1}\sum_{k=1}^n Y_k^2)^{1/2}}\rightarrow N(0,1)$$

Answer 1

We know that because $Y_{i}$ are iid $N(0,1)$ random variables then their sum $Z_{n}=\sum_{i=1}^{n}Y_{i}^{2}$ follows a Chi Squared distribution $X^{2}_{n}$ with $n$ degrees of freedom.

Also, the $\sqrt{Z_{n}}$ follows a Chi distribution $X_{n}$ with $n$ degrees of freedom.

Moreover, it is known that a standard normal random variable $Y_{1}$ divided by $\frac{\sqrt{Z_{n}}}{\sqrt{n}}$ follows a Student-t distribution with $n$ degrees of freedom. Proof of that argument can be found here A normal divided by the $\sqrt{\chi^2(s)/s}$ gives you a t-distribution -- proof.

So, we know what distribution follows the random variable $\frac{Y_{1}}{n^{-1}\sum_{i=1}^{n}Y_{i}^{2}}=\frac{Y_{1}}{\sqrt{Z_{n}}/\sqrt{n}}\sim$ Student-t(n)

Then it is sufficient, I think, to prove that $lim_{n\rightarrow \infty}f_{n}(t)=N(0,1)$ where $f_{n}(t)$ is the density of the Student-t(n) distribution.

For that kind of convergence, you can find the proofs here https://math.stackexchange.com/questions/3240536/convergence-of-students-t-distribution-to-a-standard-normal, here https://math.stackexchange.com/questions/2246154/show-t-distribution-is-close-to-normal-distribution-when-df-is-large, and here Limit of $t$-distribution as $n$ goes to infinity