Past exam of the mathematics course of the University of Cambridge / 2009 / ii / Paper 3 / 25J / Solution

For events $A_n$, their limsup is the event that infinitely many occur. The first Borel-Cantelli lemma states that ${\sum }_n\mathbb{P}(A_n)<\infty$ implies $\mathbb{P}(\mathrm{limsup}{A}_n)=0$, without independence. Indeedand intersecting these decreasing tail-unions proves the claim.

The second Borel-Cantelli lemma states that independent events with ${\sum }_n\mathbb{P}(A_n)=\infty$ satisfy $\mathbb{P}(\mathrm{limsup}{A}_n)=1$. For each fixed $N$,Thus the probability that no event occurs after $N$ is zero; take the countable union over $N$ to exclude finitely many occurrences.

As printed, the requested real logarithm is undefined: independent Cauchy random variables are negative with probability $1/2$, and the second Borel-Cantelli lemma shows that this happens infinitely often almost surely. There is therefore no real-valued sequence $\log X_n/\log n$ as stated. The natural corrected expression uses $\log |X_n|$ (or takes the limsup only over indices with $X_n>0$).

For the absolute-value interpretation,For every $q>1$ the events $|X_n|>n^q$ are summable and hence occur only finitely often; for $0<q\le 1$ they are independent with divergent probability sum and occur infinitely often. Applying the two Borel-Cantelli lemmas to a countable sequence of $q$ approaching one from above and below givesThe one-sided tail $\mathbb{P}(X_n>t)\sim 1/(\pi t)$ gives the same constant $1$ for the positive-subsequence interpretation. It does not repair the undefined logarithms at the remaining indices unless that interpretation is explicitly adopted.