The Decomposition-Separation Theorem for Finite Nonhomogeneous Markov Chains and Related Problems

Abstract

Let M be a finite set, P be a stochastic matrix and U = f(Zn)g be the family of all finite Markov chains (MC) (Zn) defined by M;P, and all possible initial distributions. The behavior of a MC (Zn) is a classical result of Probability Theory derived in the 30's by A. Kolmogorov and W. Doeblin. If a stochastic matrix P is replaced by a sequence of stochastic matrices (Pn) and transitions at moment n are defined by Pn, then U becomes a family of nonhomogeneous MCs. There are numerous results concerning the behavior of such MCs given some specific properties of the sequence (Pn): But what if there are no assumptions about sequence (Pn)? Is it possible to say something about the behavior of the family U ? The surprising answer to this question is Yes. Such behavior is described by a theorem which we call a Decomposition- Separation (DS) Theorem, and which was initiated by a small paper of A. N. Kolmogorov (1936) and formulated and proved in a few stages in a series of papers including: D. Blackwell (1945), H. Cohn (1971, 1989) and I. Sonin (1987, 1991, 1996).