Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 2 / 27K / d / Solution

Under the transience assumption, the number of returns to zero is geometrically distributed and hence finite almost surely. The final excursion is a pure ascent through all sufficiently large states. Its holding times are independent exponential random variables with meansThe nonexplosion criterion for a pure birth process says that their sum is finite almost surely exactly when the sum of their means is finite. Thus