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Weak convergence of conditioned birth-death processes in discrete time

  • Pauline Schrijner (a1) and Erik A. Van Doorn (a2)

Abstract

We consider a discrete-time birth-death process on the non-negative integers with −1 as an absorbing state and study the limiting behaviour as n → ∞ of the process conditioned on non-absorption until time n. By proving that a condition recently proposed by Martinez and Vares is vacuously true, we establish that the conditioned process is always weakly convergent when all self-transition probabilities are zero. In the aperiodic case we obtain a necessary and sufficient condition for weak convergence.

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Corresponding author

Postal address: Department of Mathematics, University of Durham, Science Laboratories, South Road, Durham DH1 3LE, UK. E-mail address: pauline.schrijner@durham.ac.uk
∗∗ Postal address: Faculty of Applied Mathematics, University of Twente, PO Box 217, 7500 AE Enschede, The Netherlands. E-mail address: doorn@math.utwente.nl

References

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[1] Karlin, S. and Mcgregor, J. L. (1959) Random walks. Illinois J. Math. 3, 6681.
[2] Martínez, S. and Vares, M. E. (1995) A Markov chain associated with the minimal quasi-stationary distribution of birth-death chains. J. Appl. Prob. 32, 2538.
[3] Roberts, G. O. and Jacka, S. D. (1994) Weak convergence of conditioned birth and death processes. J. Appl. Prob. 31, 90100.
[4] Roberts, G. O., Jacka, S. D. and Pollett, P. K. (1997) Non-explosivity of limits of conditioned birth and death processes. J. Appl. Prob. 34, 3545.
[5] Van Doorn, E. A. and Schrijner, P. (1992) Random walk polynomials and random walk measures. J. Comput. Appl. Prob. 49, 289296.
[6] Van Doorn, E. A. and Schrijner, P. (1995) Geometric ergodicity and quasi-stationarity in discrete-time birth-death processes. J. Austral. Math. Soc. B 37, 121144.
[7] Van Doorn, E. A. and Schrijner, P. (1995) Ratio limits and limiting conditional distributions for discrete-time birth-death processes. J. Math. Anal. Appl. 190, 263284.

Keywords

MSC classification

Weak convergence of conditioned birth-death processes in discrete time

  • Pauline Schrijner (a1) and Erik A. Van Doorn (a2)

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