On random convex chains, orthogonal polynomials, PF sequences and probabilistic limit theorems

TL;DR

This paper studies the convex hulls formed by random points in a triangle, revealing their probability generating functions have real roots and their vertex count distributions form PF sequences, with implications for probabilistic limit theorems.

Contribution

It establishes a three-term recursion for the generating function of the number of vertices and links it to orthogonal polynomials, showing the vertex count distribution is a PF sequence.

Findings

The probability generating function has exactly n real roots in (-∞,0].

The vertex count distribution forms a Polya frequency sequence.

Probabilistic consequences of these properties are discussed.

Abstract

Let $T$ be the triangle in the plane with vertices $(0,0)$, $(0,1)$ and $(0,1)$. The convex hull of $(0,1)$, $(1,0)$ and $n$ independent random points uniformly distributed in $T$ is the random convex chain $T_n$. A three-term recursion for the probability generating function $G_n$ of the number $f_0(T_n)$ of vertices of $T_n$ is proved. Via the link to orthogonal polynomials it is shown that $G_n$ has precisely $n$ distinct real roots in $(-\infty,0]$ and that the sequence $p_k^{(n)}:=\mathbb{P}(f_0(T_n)=k)$, $k=1,\ldots,n$, is a Polya frequency (PF) sequence. A selection of probabilistic consequences of this surprising and remarkable fact are discussed in detail.