Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 201 / 1

The partial sums $S_n$ are a martingale, as are $S_n^2-n$. Since the increment has mean zero and nonzero variance, there are ${\delta }_{+},{\delta }_{-}>0$ with $\mathbb{P}(X_1\ge {\delta }_{+})>0$ and $\mathbb{P}(X_1\le -{\delta }_{-})>0$. From any point in $(-a,b)$, a sufficiently long run of either kind exits the interval. Independence in consecutive blocks therefore bounds $\mathbb{P}(T>km)$ by a geometric sequence. In particular, $T<\infty$ almost surely and $\mathbb{E}T<\infty$.

Apply the optional stopping theorem first to $T\wedge n$. Since the increments are bounded and $\mathbb{E}T<\infty$, the stopped variables are uniformly integrable and passage to the limit givesApplying the same argument to $S_n^2-n$, using $|S_T|\le \max \{a,b\}+c$, gives

On upper exit, $b\le S_T\le b+c$; on lower exit, $-a-c\le S_T\le -a$. With $p=\mathbb{P}(S_T\ge b)$ and $\mathbb{E}{S}_T=0$,Solving gives

Moreover $(S_T+a)(S_T-b)\ge 0$, so $\mathbb{E}{S}_T^2\ge ab$, which is stronger than the requested lower bound. Also $S_T\in [-a-c,b+c]$, and henceTaking expectations and using $\mathbb{E}{S}_T=0$ gives $\mathbb{E}{S}_T^2\le (a+c)(b+c)$, stronger than the requested upper bound. Sincethe two stated estimates follow from part a.