Approximate aggregation of the states of a Markov chain with weak connections (Q1110920)

A method is proposed for the approximate calculation of limiting probabilities of aggregated states of an \(\epsilon\)-disturbed Markov chain. The author assumes the existence of 0-th approximation corresponding to the scheme described by \textit{J. G. Kemeny} and \textit{J. L. Snell} in ``Finite Markov chains''. (1960; Zbl 0089.137), for the third 1983 edition see Zbl 0537.60061, and that the limiting probabilities exist for it. The original chain is decomposed into m independent subsystems in the sense that the transition probabilities are determined only by the states of subsystems to each of which corresponds a simple homogeneous Markov chain. A Markov chain with weak connections is defined as a chain with the same set of states and transition matrix \(P=A+\epsilon B\), where A is the transition matrix of the generating system, \(\epsilon\) a small parameter and \(B=\| b_{ij}\|^ N_ 1\), \(\sum^{N}_{j=1}b_{ij}=0\) \((i=1,...,N).\) Approximate aggregation is understood as the approximate calculation of limiting probabilities of aggregated states. The accuracy of approximation is estimated, and the conditions of existence of an approximate solution are also found.