Ranking-based rich-get-richer dynamics

We consider a discrete-time, $d$-dimensional Markov processes $X_n$, for which the distribution of the future increments depends only on the relative ranking (descending order) of its components. We endow the process with a rich-get-richer assumption and show that it is enough to guarantee almost sure convergence of $X_n$ / $n$. Under mild assumptions, we characterize the possible limits via an easy to check criterion. The presented framework generalizes ranking-based Polya urns and simplifies the identification of the support of the limit.