 Solution

Let independent Bernoulli random variables $Y_n$ have success probabilities $2^{-n}$, for $n\ge 1$, and use their natural filtration. DefineEach increment is integrable, finite, and independent of the previous filtration, with expectationThus $X$ is a martingale. The increments have no common finite bound, since a success produces a jump of size $2^n-1$.

Since ${\sum }_n\mathbb{P}(Y_n=1)<\infty$, the First Borel-Cantelli lemma says that only finitely many successes occur almost surely. Consequentlyfor all sufficiently large $n$, with the random sum finite almost surely. Therefore $X_n\to +\infty$ almost surely, so neither finite convergence nor two-sided oscillation occurs. This rare-jump martingale divergence example gives probability zero, rather than one, for the union in (i).