Rare-jump martingale divergence ($X_n={\sum }_{k=1}^n(1-2^k{Y}_k)$)

A martingale can diverge to $+\infty$ almost surely when its increments are finite but not uniformly bounded. For independent Bernoulli random variables with $\mathbb{P}(Y_n=1)=2^{-n}$,has centered integrable increments. The First Borel-Cantelli lemma makes the rare negative jumps occur only finitely often, after which every increment equals one. The expectation remains zero at every finite time despite this pathwise divergence.