Consensus algorithms and the decomposition-separation theorem

Convergence properties of time inhomogeneous Markov chain based discrete time linear consensus algorithms are analyzed. Provided that a so-called infinite jet flow property is satisfied by the underlying chains, necessary conditions for both consensus and multiple consensus are established. A recent extension by Sonin of the classical Kolmogorov-Doeblin decomposition-separation for homogeneous Markov chains to the non homogeneous case, is then employed to show that the obtained necessary conditions are also sufficient when the chain is of class P*, as defined by Touri and Nedič. It is also shown that Sonin's theorem leads to a rediscovery and generalization of most of the existing related consensus results in the literature.