Matrix majorization in large samples

Understand

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.