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 discuss...

When the process underlying DNA substitutions varies across evolutionary history, some standard Markov models underlying phylogenetic methods are mathematically inconsistent. The most prominent example is the general time-reversible model (GTR) together with some, but not all, of its submodels. To rectify this deficiency, nonhomogeneous Lie Markov models have been identified as the class of mod...

A class of nonhomogeneous Markov systems with hierarchic state space working in diierent scales of time (slow and fast) that are adequate mathematical models at the analysis and modelling of various classes of computing systems and networks of a complex stochastic structure is studied. Models of asymptotic decreasing dimension and enlargement (merging) of state space are considered. Application...