Past exam of the martingales and financial mathematics course of the University of Oxford / 2010 / A / 2 / b

For any $\lambda >0$, the variable $Z=|X_1|/\lambda$ has infinite expectation. Part (a)(ii) therefore forces ${\sum }_n\mathbb{P}(|X_1|\ge \lambda n)=\infty$. Identical distributions give

For each positive integer $m$, the events $\{|X_n|\ge mn\}$ are independent and have divergent total probability by part (i). The Borel-Cantelli second lemma makes them occur infinitely often. Taking the countable intersection over $m$ givesIf ${\mathrm{limsup}}_n|S_n|/n$ were finite along any such sample path, then for some finite path-dependent $C$, $|S_n|\le Cn$ eventually. But $X_n=S_n-S_{n-1}$ would give $|X_n|/n\le C+C(n-1)/n\le 2C$ eventually, a contradiction. ThereforeThis is the divergence of normalized iid partial sums without a first absolute moment.