A birth-death process on the nonnegative integers is considered with birth rates \(\lambda_i\) and death rates \(\mu_i\), strictly positive for \(i\geq 1\) and \(\mu_0=0\). Let \(N(t)\) denote the process with \(\lambda_0>0\), let \(T(i)\) denote the first passage time from \(i\geq 1\) to 0, and let \(X(t)\) denote the absorbing birth-death process where \(\lambda_0=0\). The authors investigate the existence of the quasi- stationary distribution \(\displaystyle\lim_{t\to \infty} \text{Pr}\{N(t)=j \mid T(i)>t\}\) which is equivalent to the limiting distribution \(\displaystyle\lim_{t\to \infty} \text{Pr}\{X(t)=j \mid X(t)>0\}\).
Using methodology introduced by P. Good [J. Aust. Math. Soc. 8, 716–722 (1968; Zbl 0185.46002)] and J. A. Cavender [Adv. Appl. Probab. 10, 570–586 (1978; Zbl 0381.60068)], the authors show results on the convergence of finite quasi-stationary distributions and they give a stochastic bound for an infinite quasi- stationary distribution.