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

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.

For each integer $m$, stop on first crossing below $-m$. Bounded increments ensure the stopped martingale is bounded below by $-m-C$; after adding $m+C$ it is a nonnegative supermartingale and therefore converges. Thus on the event that $(X_n)$ is bounded below, it converges finitely. Applying the same argument to $-X_n$ shows that boundedness above also forces convergence. Outside the finite-limit event the path is therefore unbounded in both directions, so its limsup is $+\infty$ and its liminf is $-\infty$. Hence $\mathbb{P}(A\cup B)=1$.

Let $Z$ be a fair Bernoulli variable measurable at time zero and let $(S_n)$ be an independent simple symmetric random walk. Then $X_n=Z{S}_n$ is a martingale with increments bounded by one. On $\{Z=0\}$ it converges to zero, while on $\{Z=1\}$ the recurrence of the simple symmetric random walk gives limsup $+\infty$ and liminf $-\infty$. Thus $\mathbb{P}(A)=\mathbb{P}(B)=1/2$.

Set $p_n=\mathbb{P}(B_n\mid {\mathcal{F}}_{n-1})$, $S_n={\sum }_{j\le n}{1}_{B_j}$, and $A_n={\sum }_{j\le n}{p}_j$. Then $M_n=S_n-A_n$ is a martingale with bounded increments and conditional variance at most $p_n$. On $\{A_{\infty }<\infty \}$, localization and the $L^2$ martingale convergence theorem make $M_n$ converge, so the integer-valued increasing sequence $S_n$ is finite. On $\{A_{\infty }=\infty \}$, applying martingale convergence toand Kronecker lemma gives $M_n/A_n\to 0$. Hence $S_n/A_n\to 1$ and $S_n\to \infty$. This is the Conditional Borel-Cantelli lemma, and proves the two events equal almost surely.