Positive Reinforced Generalized Time-Dependent Pólya Urns via Stochastic Approximation

Wioletta M. Ruszel*, Debleena Thacker

*Corresponding author for this work

Research output: Contribution to journalArticleAcademicpeer-review

Abstract

Consider a generalized time-dependent Pólya urn process defined as follows. Let d∈N be the number of urns/colors. At each time n, we distribute σn balls randomly to the d urns, proportionally to f, where f is a valid reinforcement function. We consider a general class of positive reinforcement functions R assuming some monotonicity and growth condition. The class R includes convex functions and the classical case f(x)=xα, α>1. The novelty of the paper lies in extending stochastic approximation techniques to the d-dimensional case and proving that eventually the process will fixate at some random urn and the other urns will not receive any balls anymore.

Original languageEnglish
Pages (from-to)2859-2885
Number of pages27
JournalJournal of Theoretical Probability
Volume37
Issue number4
Early online date4 Sept 2024
DOIs
Publication statusPublished - 2024

Bibliographical note

Publisher Copyright:
© The Author(s) 2024.

Funding

W.M.R. was supported by the NWO (Dutch Research Organization) grants OCENW.KLEIN.083 and VI.Vidi.213.112.

FundersFunder number
Nederlandse Organisatie voor Wetenschappelijk Onderzoek
Dutch Research OrganizationOCENW.KLEIN.083

    Keywords

    • 60F10
    • Dominance
    • Fixation
    • Generalized Pólya urn models
    • Positive reinforcement
    • Primary 60F05
    • Secondary 60G50
    • Stochastic approximation
    • Time-dependent Pólya urn models

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