The diminishing segment process

TL;DR

This paper studies a recursive process of intersecting segments with randomly chosen points, proving convergence in distribution of the segment radius and characterizing the limiting distribution of the center.

Contribution

It introduces a novel recursive segment process and proves the asymptotic distribution of the segment radius and center.

Findings

Segment radius scaled by n converges to an exponential distribution.

Center of the limiting segment follows an arcsine distribution.

Provides rigorous probabilistic analysis of the process.

Abstract

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.