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Stochastic ordering for birth–death processes with killing
- Part of
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- Journal:
- Journal of Applied Probability / Volume 58 / Issue 3 / September 2021
- Published online by Cambridge University Press:
- 16 September 2021, pp. 708-720
- Print publication:
- September 2021
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- Article
- Export citation
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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 
$n\ge 2$
.
