Open Access
December 2023 Ranking-based rich-get-richer processes
Pantelis P. Analytis, Alexandros Gelastopoulos, Hrvoje Stojic
Author Affiliations +
Ann. Appl. Probab. 33(6A): 4366-4394 (December 2023). DOI: 10.1214/22-AAP1921


We study a discrete-time Markov process XnRd for which the distribution of the future increments depends only on the relative ranking of its components (descending order by value). We endow the process with a rich-get-richer assumption and show that, together with a finite second moments assumption, it is enough to guarantee almost sure convergence of Xn/n. We characterize the possible limits if one is free to choose the initial state and we give a condition under which the initial state is irrelevant. Finally, we show how our framework can account for ranking-based Pólya urns and can be used to study ranking algorithms for web interfaces.

Funding Statement

Pantelis P. Analytis was supported in part through NSF Award IIS-1513692 granted to Thorsten Joachims.


We would like to thank Thorsten Joachims, Gabor Lugosi, and Murad Taqqu for their remarks in previous versions of this manuscript.


Download Citation

Pantelis P. Analytis. Alexandros Gelastopoulos. Hrvoje Stojic. "Ranking-based rich-get-richer processes." Ann. Appl. Probab. 33 (6A) 4366 - 4394, December 2023.


Received: 1 May 2021; Revised: 1 August 2022; Published: December 2023
First available in Project Euclid: 4 December 2023

MathSciNet: MR4674054
Digital Object Identifier: 10.1214/22-AAP1921

Primary: 60J05
Secondary: 60J20

Keywords: Markov process , Pólya urn , ranking , rich-get-richer

Rights: Copyright © 2023 Institute of Mathematical Statistics

Vol.33 • No. 6A • December 2023
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