Past exam of the martingales and financial mathematics course of the University of Oxford / 2011 / A / 2 / c / iii

Let $E_n=\{|X_n-1|=n\}$. These events are independent and ${\sum }_{n\ge 2}\mathbb{P}(E_n)={\sum }_{n\ge 2}1/(n\log n)=\infty$. By the Borel-Cantelli second lemma, they occur infinitely often almost surely.

If $S_n/n$ had any finite sample-path limit, thenBut on $E_{n+1}$, $|X_{n+1}|/n$ is either $1$ or $(n+2)/n$, contradicting this convergence infinitely often. Therefore the averages have no finite almost sure limit. This is a rare-jump counterexample to a strong law with convergent first-moment averages.