Rosenblatt’s Contributions to Random Walks on Compact Semigroups

Abstract

Murray Rosenblatt's interest in random walks on compact semigroups probably came from his work on representations of stationary processes as shifts of functions of independent random variables described in [11], where products of matrices were studied. His papers [12], [5], [13], [14] generalized the work of Levy [2] on random walks on the circle and the work of Kawada and ltd [4] on random walks on compact groups. In [14], he also completely characterized the structure of the limit measures for the special case of compact semigrougs of n x n stochastic matrices.To describe Rosenblatt's work in this area in more detail, we first introduce some definitions and basic facts about compact topological semigroups (see [15] and [3]).1. Every compact topological semigroup S has a minimal two sided ideal K which is called the kernel of S.2. The kernel if is a compact completely simple semigroup which has a Rees structure 1x6x7, where (a) X is a compact left-zero semigroup, Y is a compact right-zero semigroup and G is a compact group.(b) The topology on X x G x Y is the product topology.