On a property of random walk polynomials involving Christoffel functions

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.