On P\'olya-Young urn models and growth processes

TL;DR

This paper extends the theory of Pólya-Young urns, providing new limit laws, martingale structures, and connections to growth processes, combinatorial objects, and generalized models involving multiple colors and complex dynamics.

Contribution

It introduces several extensions of Pólya-Young urn models, including limit laws, martingale structures, and links to growth processes and combinatorial objects, broadening the understanding of their behavior.

Findings

Determined the limit law involving local time of noise-reinforced Bessel processes.

Established a martingale structure leading to almost-sure convergence.

Derived a central limit theorem and law of the iterated logarithm for the models.

Abstract

This work is devoted to P\'olya-Young urns, a class of periodic P\'olya urns of importance in the analysis of Young tableaux. We provide several extension of the previous results of Banderier, Marchal and Wallner [Ann. Prob. (2020)] on P\'olya-Young urns and also generalize the previously studied model. We determine the limit law of the generalized model, involving the the local time of noise-reinforced Bessel processes. We also uncover a martingale structure, which leads directly to almost-sure convergence of the random variable of interest. This allows us to add second order asymptotics by providing a central limit theorem for the martingale tail sum, as well as a law of the iterated logarithm. We also turn to random vectors and obtain the limit law of P\'olya-Young urns with multiple colors. Additionally, we introduce several growth processes and combinatorial objects, which are closely related to urn models. We define increasing trees with periodic immigration and we related the dynamics of the P\'olya-Young urns to label-based parameters in such tree families. Furthermore, we discuss a generalization of Stirling permutations and obtain a bijection to increasing trees with periodic immigration. Finally, we introduce a chinese restaurant process with competition and relate it to increasing trees, as well as P\'olya-Young urns.