Given $X_1, X_2,..., X_n$ independent random variables with $P(X_n=4^n)=P(X_n=-4^n)=1/2$. Let $S_n = X_1 + X_2 +... + X_n$. Determine for which $\epsilon>0$, if any, that the $P(|S_n|/n>\epsilon)$ does not converge to $0$. Why does this result not contradict with WLLN?
For the first part of the question, we can find that for all $\epsilon$, $P(|S_n|/n>\epsilon)$ does not converge to 0. However, I am not sure why does it not contradicting with WLLN, since $E(|X|)=4^n$ it cannot be finite... How can we proof it in this case?