Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 201 / 4 / a

The almost-sure Martingale convergence theorem states that a supermartingale $(X_n)$ satisfying ${\sup }_n\mathbb{E}[X_n^{-}]<\infty$ converges almost surely to a finite random variable.

For a nonnegative supermartingale, $X_n^{-}=0$, so the hypothesis of part (i) holds. Its almost-sure limit is finite.

Let independent increments $D_n$ equal $-1$ with probability $1-2^{-n}$ and $2^n-1$ with probability $2^{-n}$. Then $\mathbb{E}{D}_n=0$, so $X_n={\sum }_{j=1}^n{D}_j$ is a martingale. Since ${\sum }_n{2}^{-n}<\infty$, the Borel-Cantelli lemmas say that only finitely many positive jumps occur almost surely. Thereafter every increment is $-1$, and hence $X_n\to -\infty$ almost surely.