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A Matrix Approach to Nonstationary Chains

Author

Listed:
  • 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)

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.

Suggested Citation

  • Frank Harary & Benjamin Lipstein & George P. H. Styan, 1970. "A Matrix Approach to Nonstationary Chains," Operations Research, INFORMS, vol. 18(6), pages 1168-1181, December.
    • Handle: RePEc:inm:oropre:v:18:y:1970:i:6:p:1168-1181
    DOI: 10.1287/opre.18.6.1168
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    Cited by:

    1. Patrice Dion, 2017. "An Alternative to Fixed Transition Probabilities for the Projection of Interprovincial Migration in Canada," Population Research and Policy Review, Springer;Southern Demographic Association (SDA), vol. 36(6), pages 871-901, December.

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