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 language | English |
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Pages (from-to) | 2859-2885 |
Number of pages | 27 |
Journal | Journal of Theoretical Probability |
Volume | 37 |
Issue number | 4 |
Early online date | 4 Sept 2024 |
DOIs | |
Publication status | Published - 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.
Funders | Funder number |
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Nederlandse Organisatie voor Wetenschappelijk Onderzoek | |
Dutch Research Organization | OCENW.KLEIN.083 |
Keywords
- 60F10
- Dominance
- Fixation
- Generalized Pólya urn models
- Positive reinforcement
- Primary 60F05
- Secondary 60G50
- Stochastic approximation
- Time-dependent Pólya urn models