One tuple of probability vectors is more informative than another tuple when there exists a single stochastic matrix transforming the probability vectors of the first tuple into the probability vectors of the other. This is called matrix majorization. Solving an open problem raised by Mu et al, we show that if certain monotones - namely multivariate extensions of R\'{e}nyi divergences - are strictly ordered between the two tuples, then for sufficiently large $n$, there exists a stochastic matrix taking the $n$-fold Kronecker power of each input distribution to the $n$-fold Kronecker power of the corresponding output distribution. The same conditions, with non-strict ordering for the monotones, are also necessary for such matrix majorization in large samples. Our result also gives conditions for the existence of a sequence of statistical maps that asymptotically (with vanishing error) convert a single copy of each input distribution to the corresponding output distribution with the help of a catalyst that is returned unchanged. Allowing for transformation with arbitrarily small error, we find conditions that are both necessary and sufficient for such catalytic matrix majorization. We derive our results by building on a general algebraic theory of preordered semirings recently developed by one of the authors. This also allows us to recover various existing results on majorization in large samples and in the catalytic regime as well as relative majorization in a unified manner.

翻译：一个概率向量元组比另一个概率向量元组包含更多信息，当存在一个随机矩阵将前一个元组的概率向量变换为后一个元组的概率向量时，这一性质被称为矩阵优超。通过解决Mu等人提出的一个开放问题，我们证明：如果某些单调量（即Rényi散度的多变量扩展）在两个元组之间严格有序，则对于足够大的n，存在一个随机矩阵将每个输入分布的n重克罗内克幂映射到相应输出分布的n重克罗内克幂。对于大规模样本中的此类矩阵优超，这些条件（单调量非严格有序）也是必要的。我们的结果还给出了统计映射序列存在的条件，这些序列能渐近地（误差趋近于零）在催化剂（可原样返还）的辅助下，将每个输入分布的单个副本转化为相应的输出分布。考虑允许任意小误差的变换时，我们找到了此类催化矩阵优超的充分必要条件。我们通过建立在最近由一位作者提出的预序半环的一般代数理论基础上推导出这些结果。这一框架也使我们能够统一地恢复关于大规模样本中优超、催化条件下的优超以及相对优超的多种现有结论。