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Monotone infinite stochastic matrices and their augmented truncations

Author

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  • Gibson, Diana
  • Seneta, E.

Abstract

Let P be a positive-recurrent, stochastically monotone, stochastic matrix on the positive integers, with stationary vector [pi]. Let (n)P be an (n x n) stochastic matrix where 1, and (n)P is the (n x n) northwest corner truncation of P, and suppose (n)[pi] is any stationary vector of (n)P. We show that (n)[pi] --> [pi] elementwise as n --> [infinity]. One corollary is the convergence to [pi] of quasistationary distributions of the (n)P. Another is that the conditions on P itself can be relaxed to domination of P by a positive-recurrent, stochastically monotone matrix R.

Suggested Citation

  • Gibson, Diana & Seneta, E., 1987. "Monotone infinite stochastic matrices and their augmented truncations," Stochastic Processes and their Applications, Elsevier, vol. 24(2), pages 287-292, May.
    • Handle: RePEc:eee:spapps:v:24:y:1987:i:2:p:287-292
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    Cited by:

    1. Liu, Jinpeng & Liu, Yuanyuan & Zhao, Yiqiang Q., 2022. "Augmented truncation approximations to the solution of Poisson’s equation for Markov chains," Applied Mathematics and Computation, Elsevier, vol. 414(C).
    2. Cruz, Juan Alberto Rojas, 2020. "Sensitivity of the stationary distributions of denumerable Markov chains," Statistics & Probability Letters, Elsevier, vol. 166(C).
    3. Badredine Issaadi, 2020. "Weak stability bounds for approximations of invariant measures with applications to queueing," Methodology and Computing in Applied Probability, Springer, vol. 22(1), pages 371-400, March.
    4. 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.
    5. Simonot, F., 1995. "Sur l'approximation de la distribution stationnaire d'une chaîne de Markov stochastiquement monotone," Stochastic Processes and their Applications, Elsevier, vol. 56(1), pages 133-149, March.
    6. Braunsteins, Peter & Decrouez, Geoffrey & Hautphenne, Sophie, 2019. "A pathwise approach to the extinction of branching processes with countably many types," Stochastic Processes and their Applications, Elsevier, vol. 129(3), pages 713-739.
    7. Jianyu Cao & Weixin Xie, 2017. "Stability of a two-queue cyclic polling system with BMAPs under gated service and state-dependent time-limited service disciplines," Queueing Systems: Theory and Applications, Springer, vol. 85(1), pages 117-147, February.

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