Ranking-based rich-get-richer processes

Abstract

We study a discrete-time Markov process ${\mathit{X}}_{\mathit{n}}\in {\mathbb{R}}^{\mathit{d}}$ 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 ${\mathit{X}}_{\mathit{n}}/\mathit{n}$. 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.