Prove the almost surely convergence of a sequence - the law of large number

Question

let $(X_i)_{i\geq1}$ a random variable which is identically independently distributed. Also, $X_i$ is normally distributed with mean $\mu=1$, variance $\sigma^2=3$.

How to show that $\lim\limits_{n\rightarrow\infty} \dfrac{X_1+X_2+\cdots+X_n}{X_1^2+X_2^2+\cdots+X_n^2}=\dfrac 1 4 \text{ a.s.}$

I'm going through J. Jacod and P. Protter's Probability Essentials. It's my first time reading a rigorous probability theory book. I hope to understand the materials better by doing the exercises. Unfortunately, this book comes without solutions which makes me crazy.

Answer 1

If $\{X_i\}$ is i.i.d so is $\{X_i^{2}\}$. Divide numerator and denominator by $n$ and apply SLLN's to both. The almost sure limit equals $\frac {EX_1} {EX_1^{2}}$ Note that $EX_1=1$ and ${EX_1^{2}}=4$.

Answer 2

Use the linearity of expectation. The expectation of $X_i$ is $1$. What is the expectation of $X_i^2$?