The diminishing segment process

Let S(1) be the segment [-1,1], and define the segments S(n) recursively in the following manner: let S(n+1) be the intersection of S(n) and a(n+1) + S(1), where the point a(n+1) is chosen randomly on the segment S(n) with uniform distribution. For the radius r(n) of S(n) we prove that n(r(n) - 1//2) converges in distribution to an exponential law, and we also show that the centre of the limiting unit interval has arcsine distribution.