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Stochastic ordering for birth–death processes with killing

Published online by Cambridge University Press:  16 September 2021

Shoou-Ren Hsiau*
National Changhua University of Education
May-Ru Chen*
National Sun Yat-sen University
Yi-Ching Yao*
Academia Sinica
*Postal address: Department of Mathematics, National Changhua University of Education, 1 Jin-De Road, Changhua 500, Taiwan, ROC. Email address:
**Postal address: Department of Applied Mathematics, National Sun Yat-sen University, 70 Lien-hai Road, Kaohsiung 804, Taiwan, ROC. Email address:
***Postal address: Institute of Statistical Science, Academia Sinica, 128 Academia Road, Section 2, Nankang, Taipei 11529, Taiwan, ROC. Email address:


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 $n\ge 2$ .

Original Article
© The Author(s), 2021. Published by Cambridge University Press on behalf of Applied Probability Trust

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