This paper studies the majorization of high tensor powers of finitely supported probability distributions. Taking two probability distributions P and q to the n 'th and m 'th tensor power, respectively, in such a way that the power of q majorizes the power of P , we ask how large the ratio m/n can become. It is shown that the supremum of such ratios is equal to the minimal ratio of the α -Rényi entropies of P and q for α \in [0,∞]. Consideration of this ratio of tensor powers is motivated, to the author, by the resource theory of quantum entanglement, where the supremum of these ratios corresponds to the asymptotic conversion rate of bipartite pure quantum states under exact, deterministic LOCC transformations.