Positive Reinforced Generalized Time-Dependent Pólya Urns via Stochastic Approximation

Wioletta M. Ruszel () and Debleena Thacker ()
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Wioletta M. Ruszel: Utrecht University
Debleena Thacker: Durham University

Journal of Theoretical Probability, 2024, vol. 37, issue 4, 2859-2885

Abstract: Abstract Consider a generalized time-dependent Pólya urn process defined as follows. Let $$d\in \mathbb {N}$$ d ∈ N be the number of urns/colors. At each time n, we distribute $$\sigma _n$$ σ n 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 $$\mathcal {R}$$ R assuming some monotonicity and growth condition. The class $$\mathcal {R}$$ R includes convex functions and the classical case $$f(x)=x^{\alpha }$$ f ( x ) = x α , $$\alpha >1$$ 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.

Keywords: Generalized Pólya urn models; Time-dependent Pólya urn models; Positive reinforcement; Stochastic approximation; Dominance; Fixation; Primary 60F05; 60F10; Secondary 60G50 (search for similar items in EconPapers)
Date: 2024
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DOI: 10.1007/s10959-024-01366-w

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