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Stochastic ordering for birth–death processes with killing

Part of: Distribution theory - Probability Markov processes

Published online by Cambridge University Press:  16 September 2021

Shoou-Ren Hsiau ,
May-Ru Chen  and
Yi-Ching Yao
Shoou-Ren Hsiau*
Affiliation:
National Changhua University of Education
May-Ru Chen*
Affiliation:
National Sun Yat-sen University
Yi-Ching Yao*
Affiliation:
Academia Sinica
*
*Postal address: Department of Mathematics, National Changhua University of Education, 1 Jin-De Road, Changhua 500, Taiwan, ROC. Email address: srhsiau@cc.ncue.edu.tw
**Postal address: Department of Applied Mathematics, National Sun Yat-sen University, 70 Lien-hai Road, Kaohsiung 804, Taiwan, ROC. Email address: mayru@faculty.nsysu.edu.tw
***Postal address: Institute of Statistical Science, Academia Sinica, 128 Academia Road, Section 2, Nankang, Taipei 11529, Taiwan, ROC. Email address: yao@stat.sinica.edu.tw
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Abstract

We consider a birth–death process with killing where transitions from state i may go to either state $i-1$ or state $i+1$ or an absorbing state (killing). Stochastic ordering results on the killing time are derived. In particular, if the killing rate in state i is monotone in i, then the distribution of the killing time with initial state i is stochastically monotone in i. This result is a consequence of the following one for a non-negative tri-diagonal matrix M: if the row sums of M are monotone, so are the row sums of $M^n$ for all $n\ge 2$ .

Keywords

Absorbing timeuniformizable chainbirth–death process with catastrophestri-diagonal matrix.

MSC classification

Primary: 60E15: Inequalities; stochastic orderings
Secondary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 60J27: Continuous-time Markov processes on discrete state spaces
Type
Original Article
Information
Journal of Applied Probability , Volume 58 , Issue 3 , September 2021 , pp. 708 - 720
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Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Applied Probability Trust

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