Ranking-based rich-get-richer processes
Pantelis P. Analytis, Alexandros Gelastopoulos, Hrvoje Stojić · The Annals of Applied Probability · 2023
We study a discrete-time Markov process Xn∈Rd 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.