Extinction probability in a birth-death process with killing

Abstract. We study birth-death processes on the non-negative integers where {1, 2,...} is an irreducible class and 0 an absorbing state, with the additional feature that a transition to state 0 may occur from any state. We give a condition for absorption (extinction) to be certain and obtain the eventual absorption probabilities when absorption is not certain. We also study the rate of convergence as t→ ∞ of the probability of absorption at time t, and relate it to the common rate of convergence of the transition probabilities which do not involve state 0. Finally, we derive upper and lower bounds for the probability of absorption at time t by applying a technique which involves the logarithmic norm of an appropriately defined operator