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={(Z_n)} be the family of all finite Markov chains (MC) (Z_n) defined by M, P, and all possible initial distributions. The behavior of a MC (Z_n) is a classical result of probability theory derived in the 1930s by A. N. Kolmogorov and W. Doeblin. If a stochastic matrix P is replaced by a sequence of stochastic matrices (P_n) and transitions at moment n are defined by P_n, 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 (P_n). But what if there are no assumptions about sequence (P_n)? 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).