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HITTING TIME DISTRIBUTIONS FOR BIRTH–DEATH PROCESSES WITH BILATERAL ABSORBING BOUNDARIES

Published online by Cambridge University Press:  13 September 2016

Yong-Hua Mao  and
Chi Zhang
Yong-Hua Mao
Affiliation:
School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875, People's Republic of China
Chi Zhang
Affiliation:
College of Mathematical Sciences, College of Information Science and Engineering Ocean University of China, Qingdao 266100, People's Republic of China E-mail: [email protected]
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Abstract

For the birth–death process on a finite state space with bilateral boundaries, we give a simpler derivation of the hitting time distributions by h-transform and φ-transform. These transforms can then be used to construct a quick derivation of the hitting time distributions of the minimal birth–death process on a denumerable state space with exit/regular boundaries.

Keywords

stochastic modelling
Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 31 , Issue 3 , July 2017 , pp. 345 - 356
Copyright
Copyright © Cambridge University Press 2016 

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References

1. Anderson, W. (1991). Continuous–time Markov chains. New York: Springer–Verlag.Google Scholar
2. Chen, M.F. (2004). From Markov chains to non-equilibrium particle systems, 2nd ed. Singapore: Word Scientific.Google Scholar
3. Chen, M.F. (2010). Speed of stability for birth-death processes. Frontiers of Mathematics in China 5: 379–515.Google Scholar
4. Diaconis, P. & Miclo, L. (2009). On times to quasi-stationarity for birth and death processes. Journal of Theoretical Probability 22: 558–586.Google Scholar
5. Diaconis, P. & Miclo, L. (2014). On quantitative convergence to quasi-stationarity. arXiv: 1406.1805.Google Scholar
6. Diaconis, P. & L, Miclo. (2015). Estimates on the amplitude of the first Dirichlet eigenvector in discrete frameworks. Science China Mathematics 59: 205–226.Google Scholar
7. van Doorn, E.A. (2015). An orthogonal-polynomial approach to first-hitting times of birth-death processes. Journal of Theoretical Probability DOI: 10.1007/s10959-015-0659-z.Google Scholar
8. Feller, W. (1959). The birth and death processes as diffusion processes. Journal des Mathématiques Pures et Appliquées 38: 301–345.Google Scholar
9. Fill, J.A. (2009). The passage time distribution for a birth-and-death chain: strong stationary duality gives a first stochastic proof. Journal of Theoretical Probability 22: 543–557.CrossRefGoogle Scholar
10. Gong, Y., Mao, Y.H., & Zhang, C. (2012). Hitting time Distributions for denumerable birth and death processes. Journal of Theoretical Probability 25: 950–980.CrossRefGoogle Scholar
11. Karlin, S. & McGregor, J. (1959). Coincidence properties of birth and death processes. Pacific Journal of Mathematics 9: 1109–1140.Google Scholar
12. Kato, T. (1966). Perturbation theory for linear operators. Berlin: Springer-Verlag.Google Scholar
13. Keilson, J. (1979). Markov chain models–rarity and exponentiality . Applied Mathematical Sciences. vol. 28. New York: Springer.Google Scholar
14. Mao, Y.-H. (2004). The eigentime identity for continuous-time ergodic Markov chains. Journal of Applied Probability 41: 1071–1080.Google Scholar
15. Micchelli, C.A. & Willoughby, R.A. (1979). On functions which preserve the class of Stieltjes matrices. Linear Algebra and its Application 23: 141–156.Google Scholar