On a property of random walk polynomials involving Christoffel functions

Erik Alexander van Doorn (Corresponding Author), Ryszard Szwarc

    Research output: Contribution to journalArticleAcademicpeer-review

    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.
    Original languageEnglish
    Pages (from-to)85-103
    Number of pages19
    JournalJournal of mathematical analysis and applications
    Volume477
    Issue number1
    Early online date12 Apr 2019
    DOIs
    Publication statusPublished - 1 Sep 2019

    Keywords

    • (Asymptotic) period
    • (Asymptotic) aperiodicity
    • Birth-death process
    • Random walk measure
    • Ratio limit
    • Transition probability

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