Weak convergence in d x d bistochastic matrices and other semigroups

Abstract

In the first part of this dissertation, we look into the special semigroup of bistochastic matrices and show that the following problem can be solved completely: Given a probability measure $\mu$ on the Borel subsets of the compact semigroup of $d$ x $d$ bistochastic matrices (with usual topology and matrix multiplication), how we can decide whether the sequence ($\mu\sp{n}$) converges weakly or not, and in case of convergence, what the limiting measure is.

In the second part of this dissertation, we discuss various necessary and sufficient conditions for a sequence of non-identical distributions to be composition convergent when S is a non-abelian semigroup. We also extend some Maksimov's results to compact semigroups.