A Geometric Approach to Ergodic Non-homogeneous Markov Chains

Inspired by the recent work of Daubechies and Lagarias on a set of matrices with convergent innnite products, we establish a new geometric approach to the classical problem of (weakly) ergodic non-homogeneous Markov chains. The existing key inequalities (related to the Hajnal inequality) in the literature are uniied in this geometric picture. A more general inequality is established. Important quantities introduced by various authors are easily interpreted. A quantitative connection is established between the classical work of Hajnal and the more recent one of Daubechies and Lagarias. 1. Overview In this paper, we restrict ourselves to Markov chains on nite states. Section 2 and 3 together serve as an introductory section, where the historical background and recent development are brieey reviewed, and our motivation and new results are introduced. In Section 2, we review the literature of ergodic non-homogeneous Markov chains 1, 2, 4, 5, 8, 10]. The basic meaning of ergodicity is explained. Section 3 displays the eeorts of several authors to characterize the conditions for ergodicity. Hajnal's inequality 5] is singled out for its signiicant role in this course. Our most general inequality is introduced. New concepts such as-Markov chains and LCP (left-convergent-products, Daubechies and Lagarias 2]) are deened. We also claim that the projected jointed spectral radius of Daubechies and Lagarias can be characterized in a new way. Section 4 shows the motivation of our geometric approach by working out the simple example of Cantorian (non-homogeneous) Markov chains. It