A Matrix Approach to Nonstationary Chains

Frank Harary, Benjamin Lipstein and George P. H. Styan
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Frank Harary: University of Michigan, Ann Arbor, Michigan
Benjamin Lipstein: Sullivan, Stauffer, Colwell & Bayles, Inc., New York, New York
George P. H. Styan: McGill University, Montreal, Quebec, Canada

Operations Research, 1970, vol. 18, issue 6, 1168-1181

Abstract: A finite discrete nonstationary Markov chain is completely characterized (after the initial probability distribution has taken effect) by its time sequence of transition probability matrices P i . The i th causative matrix C i is defined as the product P i −1 (if it exists) times P i +1 . Thus, the causative matrices are analogous to derivatives in calculus as an indication of rate of change. The eigenvalues and eigenvectors of a constant causative matrix C have been found useful in their connection with the tendency of the chain to be convergent or divergent. Results for two-state chains are presented in some detail. A comprehensive bibliography of papers on non-stationary chains is included.

Date: 1970
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