Limit Theorems: Stochastic Matrices, Ergodic Markov Chains, and Measures on Semigroups

Abstract

This chapter reviews problems on ergodicity of non-homogeneous and homogeneous Markov chains to apply several of the results on measures on semi-groups. The chapter considers limit theorems for probability measures on stochastic matrices by first showing how these problems come up naturally in studying random walks on simple geometric figures on the plane. The abstract theory is reviewed. Completely simple semi-groups are considered to study limit theorems for convolution products of probability measures, including generalizations of the well-known Paul Levy theorem on the equivalence of convergence in probability, convergence with probability one, and convergence in distribution for sums of independent random variables. An important class of semi-groups that have been widely studied by semi-groupists and probabilists is the class of completely simple semi-groups. A semi-group is called completely simple if it is simple and contains a primitive idempotent. Various theorems are proven in the chapter. Convergence problems for products of stochastic matrices naturally come up in various contexts in biology, economics, and various other applications. The chapter discusses the problem of tendency to consensus in an information exchanging operation.