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Abstract: Let $(P_{n}), n=0,1,2,...$ be a sequence of finite stochastic matrices of the same size $N\times N$. How does the product $\prod_{i=k}^nP_i, 0\leq k$, behave, when $n\rightarrow \infty$, if there are no assumptions about sequence the $(P_{n})$? In probabilistic terms this question is equivalent to a question about the asymptotic behaviour of a family of nonhomogeneous Markov chains defined by a sequence $(P_{n})$ and all possible initial distributions on a finite set $S, |S|=N$ with an initial time point $k$.
The surprising answer to this question is given by a fundamental theorem, which we call the Decomposition-Separation (DS)\ Theorem. This theorem was initiated by a small paper of A. N. Kolmogorov (1936) and formulated and proved in a few stages in a series of papers: D. Blackwell (1945), H. Cohn (1971,..., 1989) and I. Sonin (1987, 1991,..., 2008).
The DS Theorem has a simple deterministic interpretation in terms of the behaviour of the simplest model of an irreversible process. Such a model can be defined, for
example, by a finite number of cups filled with a solution (e.g. tea) having
possibly different concentrations and exchanging this solution at
discrete times. This can also be considered in terms of the evolution of a
systems of particles, which connects this theorem with a broad area called Consensus Algorithms.
Some new results will also be presented, but generally the DS
Theorem leaves many open problems and probably leads to generalizations in other fields of mathematics besides probability theory.
Among other topics that will be briefly mentioned are: The Markov Chain Tree Theorem and the question: "What does it mean that two events are independent?"