Past exam of the mathematics course of the University of Cambridge / 2005 / iii / Paper 12 / 6 / b / Solution

The first right endpoint is the smallest of $n$ independent pair maxima. A maximum exceeds $r$ with probability $1-r^2$, so for $0\le x\le \sqrt{n}$,Taking the limit gives the requested, correctly normalized resultThus $\sqrt{n}{R}_1$ converges to a continuous nonnegative limit $W$ with survival function $\mathrm{Pr}(W\ge x)=e^{-x^2}$. In particular the scaled first right endpoint is tight. This calculation uses maxima of pairs, not the first order statistic of all $2n$ individual endpoints, which would have a different scale. The converted TeX loses the formula in this part; the displayed limit has been recovered directly from the PDF.