Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 1 / 27L (Probability and Measure)

The statements and proof are in Borel-Cantelli lemmas. For the first,
$\mathbb{P}({\bigcup }_{n\ge m}{A}_n)\le {\sum }_{n\ge m}\mathbb{P}(A_n)\to 0$. For independent events in the divergent case,and then let $m\to \infty$.

The characteristic function of $S_n/n$ isThus every $S_n/n$ already has the standard Cauchy law, and convergence in law follows from characteristic-function convergence theorem.

If $S_n/n$ converged almost surely to a finite limit, thenBut $\mathbb{P}(X_n/n>1)=\mathbb{P}(X_1>n)\sim 1/(\pi n)$. These independent-event probabilities have divergent sum, so the second Borel-Cantelli lemmas says $X_n/n>1$ infinitely often almost surely, a contradiction.