Solution

Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 34 / 2 / c / Solution

{c}
{title2=$\limsup_{n\to\infty}X_n^2/\log n=2$}

For independent <standard normal random variables> $X_n$, the displayed <limit superior> equals two <almost surely>. The <Mills ratio> asymptotic gives, for $a>0$,
$$
\mathbb P(X_n^2>a\log n)\sim\sqrt{\frac{2}{\pi a}}\frac{n^{-a/2}}{\sqrt{\log n}}.
$$
The series converges for $a>2$ and diverges for $0<a\leq2$. The two <Borel-Cantelli lemmas>, applied at countably many thresholds approaching two, give the result. In particular, the value two is also a threshold exceeded infinitely often, despite being the almost-sure <limit superior>.

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 Incoming links: Gaussian sample limsup at logarithmic scale