  Convergence properties of time inhomogeneous Markov chain based discrete and
continuous 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 recenet extension by Sonin of the classical Kolmogorov-Doeblin
decomposition-separation for homogeneous Markov chains to the inhomogeneous
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 Nedic. 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.
