A note on the extinction rates of some birth–death processes

Abstract

In this note we are concerned with the rate of extinction of certain continuous-time birth-death processes on the positive integers with absorption at 0. The class we deal with includes birth-death processes with mean holding time h(i) at i such that h (i)∼ i^–α as i →∞, 0 ≦ α< 1. In general, our result estimates to within a constant multiple the probability of non-extinction by time t. For h(i)∼ i^–α, the result states that the probability of non-extinction is of order t^−1/(2-α) We give an application to interacting particle systems.