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

Fingerprint

Random walk
Polynomials
Polynomial
Birth-death Process
Equivalence
Regularity Conditions
Orthogonal Polynomials
Discrete-time

Keywords

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

Cite this

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On a property of random walk polynomials involving Christoffel functions. / van Doorn, Erik Alexander (Corresponding Author); Szwarc, Ryszard.

In: Journal of mathematical analysis and applications, Vol. 477, No. 1, 01.09.2019, p. 85-103.

Research output: Contribution to journalArticleAcademicpeer-review

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