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Censoring, Factorizations, and Spectral Analysis for Transition Matrices with Block-Repeating Entries

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  • Yiqiang Q. Zhao

    (Carleton University)

  • Wei Li

    (University of Toledo)

  • W. John Braun

    (University of Western Ontario)

Abstract

In this paper, we use the Markov chain censoring technique to study infinite state Markov chains whose transition matrices possess block-repeating entries. We demonstrate that a number of important probabilistic measures are invariant under censoring. Informally speaking, these measures involve first passage times or expected numbers of visits to certain levels where other levels are taboo; they are closely related to the so-called fundamental matrix of the Markov chain which is also studied here. Factorization theorems for the characteristic equation of the blocks of the transition matrix are obtained. Necessary and sufficient conditions are derived for such a Markov chain to be positive recurrent, null recurrent, or transient based either on spectral analysis, or on a property of the fundamental matrix. Explicit expressions are obtained for key probabilistic measures, including the stationary probability vector and the fundamental matrix, which could be potentially used to develop various recursive algorithms for computing these measures.

Suggested Citation

  • Yiqiang Q. Zhao & Wei Li & W. John Braun, 2003. "Censoring, Factorizations, and Spectral Analysis for Transition Matrices with Block-Repeating Entries," Methodology and Computing in Applied Probability, Springer, vol. 5(1), pages 35-58, March.
    • Handle: RePEc:spr:metcap:v:5:y:2003:i:1:d:10.1023_a:1024125320911
    DOI: 10.1023/A:1024125320911
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    References listed on IDEAS

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    1. A. M. Dukhovny, 1990. "Markov chains with quasitoeplitz transition matrix: applications," International Journal of Stochastic Analysis, Hindawi, vol. 3, pages 1-12, January.
    2. Alexander M. Dukhovny, 1989. "Markov chains with quasitoeplitz transition matrix," International Journal of Stochastic Analysis, Hindawi, vol. 2, pages 1-12, January.
    3. Yiqiang Q. Zhao & W. John Braun & Wei Li, 1999. "Northwest corner and banded matrix approximations to a Markov chain," Naval Research Logistics (NRL), John Wiley & Sons, vol. 46(2), pages 187-197, March.
    4. Alexander M. Dukhovny, 1989. "Markov chains with quasitoeplitz transition matrix: first zero hitting," International Journal of Stochastic Analysis, Hindawi, vol. 2, pages 1-12, January.
    5. Winfried K. Grassmann & Daniel P. Heyman, 1993. "Computation of Steady-State Probabilities for Infinite-State Markov Chains with Repeating Rows," INFORMS Journal on Computing, INFORMS, vol. 5(3), pages 292-303, August.
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    Cited by:

    1. Masakiyo Miyazawa, 2011. "Light tail asymptotics in multidimensional reflecting processes for queueing networks," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 19(2), pages 233-299, December.
    2. Masuyama, Hiroyuki, 2011. "Subexponential asymptotics of the stationary distributions of M/G/1-type Markov chains," European Journal of Operational Research, Elsevier, vol. 213(3), pages 509-516, September.
    3. Jiashan Tang & Yiqiang Zhao, 2008. "Stationary tail asymptotics of a tandem queue with feedback," Annals of Operations Research, Springer, vol. 160(1), pages 173-189, April.
    4. A. B. Dieker & M. Mandjes, 2011. "Extremes of Markov-additive Processes with One-sided Jumps, with Queueing Applications," Methodology and Computing in Applied Probability, Springer, vol. 13(2), pages 221-267, June.

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