  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.
