Finding Limiting Distribution

Question

Let $X_i \sim N(0,1)$ for $0 \leq i \leq n$ be i.i.d. standard normal distributions.

Define $Y_n := \frac{\sqrt{n} X_0}{\sqrt{(\sum_{i=1}^n X_i^2)}}$

Find the limiting distribution function for $Y_n$.

Answer 1

$\sum_{i=1}^n X_i^2\sim\chi^2_n$ (as it is the sum of $n$ independent $N(0,1)$ variables) and is independent of $X_0$. Hence $$\frac{X_0}{\sqrt{\frac{\sum_{i=1}^n X_i^2}{n}}}\sim t_{n}$$ the Student's $t$ distribution with $n$ degrees of freedom.