Positive Reinforced Generalized Time-Dependent Pólya Urns via Stochastic Approximation

Ruszel, Wioletta M.; Thacker, Debleena

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.

Citation

Ruszel, W. M., & Thacker, D. (2024). Positive Reinforced Generalized Time-Dependent Pólya Urns via Stochastic Approximation. Journal of Theoretical Probability, 37(4), 2859-2885. https://doi.org/10.1007/s10959-024-01366-w

• Journal Article Type: Article
• Acceptance Date: Jul 31, 2024
• Online Publication Date: Sep 4, 2024
• Publication Date: Sep 4, 2024
• Deposit Date: Oct 14, 2024
• Publicly Available Date: Oct 14, 2024
• Journal: Journal of Theoretical Probability
• Print ISSN: 0894-9840
• Electronic ISSN: 1572-9230
• Publisher: Springer
• Peer Reviewed: Peer Reviewed
• Volume: 37
• Issue: 4
• Pages: 2859-2885
• DOI: https://doi.org/10.1007/s10959-024-01366-w
• Public URL: https://durham-repository.worktribe.com/output/2957116