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# On the relationship between µ-invariant measures and quasi-stationary distributions for continuous-time Markov chains

## Abstract

In a recent paper, van Doorn (1991) explained how quasi-stationary distributions for an absorbing birth-death process could be determined from the transition rates of the process, thus generalizing earlier work of Cavender (1978). In this paper we shall show that many of van Doorn's results can be extended to deal with an arbitrary continuous-time Markov chain over a countable state space, consisting of an irreducible class, C, and an absorbing state, 0, which is accessible from C. Some of our results are extensions of theorems proved for honest chains in Pollett and Vere-Jones (1992).

In Section 3 we prove that a probability distribution on C is a quasi-stationary distribution if and only if it is a µ-invariant measure for the transition function, P. We shall also show that if m is a quasi-stationary distribution for P, then a necessary and sufficient condition for m to be µ-invariant for Q is that P satisfies the Kolmogorov forward equations over C. When the remaining forward equations hold, the quasi-stationary distribution must satisfy a set of ‘residual equations' involving the transition rates into the absorbing state. The residual equations allow us to determine the value of µ for which the quasi-stationary distribution is µ-invariant for P. We also prove some more general results giving bounds on the values of µ for which a convergent measure can be a µ-subinvariant and then µ-invariant measure for P. The remainder of the paper is devoted to the question of when a convergent µ-subinvariant measure, m, for Q is a quasi-stationary distribution. Section 4 establishes a necessary and sufficient condition for m to be a quasi-stationary distribution for the minimal chain. In Section 5 we consider ‘single-exit' chains. We derive a necessary and sufficient condition for there to exist a process for which m is a quasi-stationary distribution. Under this condition all such processes can be specified explicitly through their resolvents. The results proved here allow us to conclude that the bounds for µ obtained in Section 3 are, in fact, tight. Finally, in Section 6, we illustrate our results by way of two examples: regular birth-death processes and a pure-birth process with absorption.

## Corresponding author

∗∗Postal address: Department of Mathematics, The University of Queensland, QLD 4072, Australia. E-mail address: pkp@maths.uq.oz.au

## Footnotes

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Present address: School of Mathematics and Statistics, Curtin University of Technology, Perth, WA6001, Australia. E-mail address: gopal@marsh.cs.curtin.edu.au

This research was funded under an Australian Research Council Grant and a University of Queensland Special Project Grant.

## References

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Chung, K. L. (1967) Markov Chains with Stationary Transition Probabilities, 2nd edn. Springer-Verlag, Berlin.
Van Doorn, E. A. (1991) Quasi-stationary distributions and convergence to quasi-stationarity of birth–death processes. Adv. Appl. Prob. 23, 683700.
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Knopp, K. (1956) Infinite Sequences and Series. Dover, New York.
Lamb, C. W. (1971) On the construction of certain transition functions. Ann. Math. Statist. 42, 439450.
Pollett, P. K. (1986) On the equivalence of µ-invariant measures for the minimal process and its q-matrix. Stoch. Proc. Appl. 22, 203221.
Pollett, P. K. (1988) Reversibility, invariance and µ-invariance. Adv. Appl. Prob. 20, 600621.
Pollett, P. K. (1991a) On the construction problem for single-exit Markov chains. Bull. Austral. Math. Soc. 43, 439450.
Pollett, P. K. (1991b) Invariant measures for Q-processes when Q is not regular. Adv. Appl. Prob. 23, 277292.
Pollett, P. K. and Vere-Jones, D. (1992) A note on evanescent processes. Austral. J. Statist. To appear.
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# On the relationship between µ-invariant measures and quasi-stationary distributions for continuous-time Markov chains

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