  We study a discrete-time Markov process $X_n\in\mathbb{R}^d$, 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 $X_n$ / $n$.
We characterize the possible limits if one is free to choose the initial state,
and give a condition under which the initial state is irrelevant. Finally, we
show how our framework can account for ranking-based P\'olya urns and can be
used to study ranking-algorithms for web interfaces.
