The diminishing segment process

• DOI: 10.1016/j.spl.2011.09.016
• Publication Date: 2011-10-02T08:18:02Z
• Authors: Gergely Ambrus; Péter Kevei; Viktor Vígh
• Keywords: Probability (math.PR); FOS: Mathematics; Mathematics - Combinatorics; Combinatorics (math.CO); 0101 mathematics; 01 natural sciences; Mathematics - Probability

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.