A note on random orthogonal polynomials on a compact interval

TL;DR

This paper investigates the asymptotic behavior of roots of random orthogonal polynomials generated from uniformly distributed moment sequences on [0,1], revealing new insights into their distribution as degree increases.

Contribution

It introduces a novel probabilistic model for orthogonal polynomials based on random moment sequences and analyzes their root asymptotics.

Findings

Roots exhibit specific asymptotic distribution patterns

Distribution of roots converges as degree increases

Provides theoretical foundation for random orthogonal polynomial analysis

Abstract

We consider a uniform distribution on the set $\mathcal{M}_k$ of moments of order $k \in \mathbb{N}$ corresponding to probability measures on the interval $[0,1]$. To each (random) vector of moments in $\mathcal{M}_{2n-1}$ we consider the corresponding uniquely determined monic (random) orthogonal polynomial of degree $n$ and study the asymptotic properties of its roots if $n \to \infty$.