  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.

Classical Analysis and ODEs;Probability