Question:
$\{X_k\}_{k=1}^{\infty}$ i.i.d.,$P(X_1=-1)=P(X_1=1)=\frac12$. $S_n=\sum\limits_{k=1}^n X_k$. Prove that:
1.$S_n$ converge to infinity almost surely,i.e. $P(\lim_{n\to+\infty}S_n=\infty)=1$;
2.$\frac{S_n}{\sqrt{n}}$ diverge almost surely.
Attempt:
1.I have no idea how to prove 'converge to infinity almost surely'.I've tried to prove it in this way:
If we can show $\forall k\geq 1,\sum\limits_{n=1}^\infty P(|S_n|\leq k)<+\infty$,then by Borel-Cantelli,$S_n$ converge to infinity almost surely(since $P(|S_n|\leq k\ i.o.)=0$).
But it turns out that the sum above is $\sum \frac{1}{\sqrt{n}}$ ,so this fails;
2.By computing the characteristic function $E[\exp(it\frac{S_n}{\sqrt{n}})]\to e^{-\frac{1}{2}t^2}$
I know that $\frac{S_n}{\sqrt n}$ converge to $N(0,1)$ in distribution.
Any answer is appreciated.Thanks in advanced.