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Quasi-Stationarity of Discrete-Time Markov Chains with Drift to Infinity

Author

Listed:
  • Pauline Coolen-Schrijner

    (University of Durham, Science Laboratories)

  • Phil Pollett

    (The University of Queensland)

Abstract

We consider a discrete-time Markov chain on the non-negative integers with drift to infinity and study the limiting behavior of the state probabilities conditioned on not having left state 0 for the last time. Using a transformation, we obtain a dual Markov chain with an absorbing state such that absorption occurs with probability 1. We prove that the state probabilities of the original chain conditioned on not having left state 0 for the last time are equal to the state probabilities of its dual conditioned on non-absorption. This allows us to establish the simultaneous existence, and then equivalence, of their limiting conditional distributions. Although a limiting conditional distribution for the dual chain is always a quasi-stationary distribution in the usual sense, a similar statement is not possible for the original chain.

Suggested Citation

  • Pauline Coolen-Schrijner & Phil Pollett, 1999. "Quasi-Stationarity of Discrete-Time Markov Chains with Drift to Infinity," Methodology and Computing in Applied Probability, Springer, vol. 1(1), pages 81-96, July.
    • Handle: RePEc:spr:metcap:v:1:y:1999:i:1:d:10.1023_a:1010018406356
    DOI: 10.1023/A:1010018406356
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