The diminishing segment process

Let Ξ0=[−1,1], and define the segments Ξn recursively in the following manner: for every n=0,1,…, let Ξn+1=Ξn∩[an+1−1,an+1+1], where the point an+1 is chosen randomly on the segment Ξn with uniform distribution. For the radius ρn of Ξn, we prove that n(ρn−1/2) converges in distribution to an exponential law, and we show that the centre of the limiting unit interval has arcsine distribution.