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The Decomposition-Separation Theorem generalizing the classical KolmogorovDoeblin results about the decomposition of finite homogeneous Markov chains to the nonhomogeneous case is presented. The ground-breaking result in this direction was given in the work of David Blackwell in 1945. The relation of this theorem with other problems in probability theory and Markov Decision Processes is discussed. Dedicated to David Blackwell in deep respect for his many wonderful mathematical

The asymptotic behaviour of a general finite nonhomogeneous Markov chain (the decomposition-separation theorem)