Hostname: page-component-89b8bd64d-n8gtw Total loading time: 0 Render date: 2026-05-08T03:06:01.780Z Has data issue: false hasContentIssue false
  • English
  • Français

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: zhangchi@ouc.edu.cn
Get access Rights & Permissions [Opens in a new window]

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

Information

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

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