# On a property of random walk polynomials involving Christoffel functions

## TL;DR

This paper explores a mathematical property linking orthogonal polynomials and birth-death processes, demonstrating the equivalence of two asymptotic properties through Christoffel functions under mild conditions.

## Contribution

It establishes a novel connection between asymptotic properties of birth-death processes and Christoffel functions of random walk polynomials, proving their equivalence.

## Key findings

- Equivalence of asymptotic aperiodicity and strong ratio limit property for normalized birth-death processes.
- A property involving Christoffel functions characterizes this equivalence.
- Proven under mild regularity conditions.

## Abstract

Discrete-time birth-death processes may or may not have certain properties known as asymptotic aperiodicity and the strong ratio limit property. In all cases known to us a suitably normalized process having one property also possesses the other, suggesting equivalence of the two properties for a normalized process. We show that equivalence may be translated into a property involving Christoffel functions for a type of orthogonal polynomials known as random walk polynomials. The prevalence of this property - and thus the equivalence of asymptotic aperiodicity and the strong ratio limit property for a normalized birth-death process - is proven under mild regularity conditions.