Constructing population processes with specified quasi-stationary distributions. (English) Zbl 1129.60070
Summary: This note is concerned with constructing time-homogeneous Markov chains on a countable state-space, including an absorbing state, such that the chain has a prescribed quasi-stationary distribution. The problem is characterized in terms of the underlying \(Q\)-matrix of the process. With no restriction on the Markov chain, it is straightforward to obtain a solution. For restricted processes this is no longer the case. We look in detail at population processes of birth-death and birth-catastrophe type, and in both cases obtain explicit constructions for Markov chains with any specified quasi-stationary distribution. The results are illustrated with examples.
Reviewer: Valentin Topchii (Omsk)
MSC:
| 60J27 | Continuous-time Markov processes on discrete state spaces |
| 60J80 | Branching processes (Galton-Watson, birth-and-death, etc.) |
References:
| [1] | Cavender J.A., Adv. Appl. Prob. 10 pp 570– (1978) · Zbl 0381.60068 · doi:10.2307/1426635 |
| [2] | Clancy D., Methodology and Computing in Applied Probability 3 pp 75– (2001) · Zbl 0971.92025 · doi:10.1023/A:1011418208496 |
| [3] | Cox D.R., Theory of Stochastic Processes (1965) · Zbl 0149.12902 |
| [4] | DOI: 10.2307/1427670 · Zbl 0736.60076 · doi:10.2307/1427670 |
| [5] | Ferrari P.A., A renewal dynamical approach. Ann. Prob. 23 pp 501– (1995) |
| [6] | Gilks W.R., Markov Chain Monte Carlo in Practice (1996) · Zbl 0832.00018 |
| [7] | Jacka S.D., J. Appl. Prob. 32 pp 902– (1995) · Zbl 0839.60069 · doi:10.2307/3215203 |
| [8] | Kingman J.F.C., Proc. London Math. Soc. 13 pp 337– (1963) · Zbl 0154.43003 · doi:10.1112/plms/s3-13.1.337 |
| [9] | Nair M.G., Adv. Appl. Prob. 25 pp 82– (1993) · Zbl 0774.60070 · doi:10.2307/1427497 |
| [10] | de Oliveira M.M., Physica A 343 pp 525– (2004) |
| [11] | Pakes A.G., Adv. Appl. Prob. 27 pp 120– (1995) · Zbl 0819.60065 · doi:10.2307/1428100 |
| [12] | Pollett P.K., Adv. Appl. Prob. 20 pp 600– (1988) · Zbl 0654.60058 · doi:10.2307/1427037 |
| [13] | Pollett P.K., J. Appl. Prob. 31 pp 897– (1994) · Zbl 0812.60060 · doi:10.2307/3215315 |
| [14] | Pollett P.K., Math. Comp. Modelling 22 pp 279– (1995) · Zbl 0835.60060 · doi:10.1016/0895-7177(95)00205-G |
| [15] | Pollett P.K., Aus. N. Z. J. Stat. 46 pp 113– (2004) · Zbl 1075.60094 · doi:10.1111/j.1467-842X.2004.00317.x |
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